A Square and A Cube Explained | Class 8 Maths Chapter Full Explanation

Added: 2026-05-12

[00:00:00]Hello friends, welcome to my channel.

[00:00:02]Today I am sharing a video on lesson one of class 8 Ganit Prakash which is a square and a cube. So let's start with the chapter. Okay, let's begin with a very interesting story. A queen named Ratna Manguri, the name is here. She left behind a treasure of precious stones. But instead of giving it to directly to her son, she created a puzzle. Now what was that puzzle? What she did? She invited her son and 99 more relatives. Here it is mentioned. You can see she invited his son and 99 relatives to a secret room in the mansion containing 100 lockers. Then what is the challenge? She invited 100 people in total to solve that challenge. Now each person is is assigned a number from 1 to 100. What they will do? Now person will person one will open every locker. every locker. Person two toggles every second locker like open, close, close, open.

[00:01:04]Person three toggles every third locker.

[00:01:07]Person four toggles every fourth locker.

[00:01:09]And this will will continue till all 100 people will have taken their turn. Now the big question is after all this, which locker will remain open? And here's the twist. If someone figures it out first, they get the entire treasure.

[00:01:27]But if everyone answers together, they must share it equally. Before the process even begins, the son already knew the answer. So how did he figure out? How did he figure out? Let's see.

[00:01:39]Let's understand in the next slide.

[00:01:42]Now to solve this puzzle, we need to understand one simple idea. A locker stays open only if it is toggled an odd number of times. Otherwise, it will be closed. The number of times the locker is toggled will be the same as the number of factors of the locker number.

[00:02:01]So the real question is how many times is each locker toggled? Let's see.

[00:02:06]Here's the trick. A locker is toggled by people whose numbers are its factors.

[00:02:14]What does that mean? Let's take a locker number six. Let's take a locker number six. The factors of six are 1 2 3 and six. These are the factors of six. So it is toggled four times. Four is even.

[00:02:28]Four is even. So it will be closed. Now observe something interesting. Factors usually come in pairs like 1 into 6 2 into 3. In total we have 1 2 3 6. So most number will have an in even number of factors because they always comes in pair. But some numbers are special. Some numbers are special like four. If you see four, we have 1 into 4 and 2 into 2.

[00:02:55]The numbers are repeated here. So, total factors are three. 1 4 and 2. 2 is repeated. Similarly, if you see 9, 1 into 9, 3 into 3, 3 is repeated. So, we have three factors 1, 3 and 9. Again, odd number of factors, odd number of factors. So, we see in some cases like 2 into 2 the number in the pairs are the same or we see 3 into 3, the number in the pairs are same. What does these numbers called? These numbers are called perfect squares.

[00:03:29]So finally only those lockers will remain open whose numbers are perfect squares.

[00:03:37]The open lockers are which one? 1 into 1, 2 into 2, 3 into 3, 4 into 4, 5 into 5 and so on.

[00:03:49]Can you guess how many lockers will remain open out of 100?

[00:03:54]Think about it. Out of 100, how many lockers will remain open?

[00:03:59]I need an answer in the comment box. So, from just a puzzle, we discovered an amazing property of numbers. Only perfect squares have an odd number of factors. In the next part, let's explore what square numbers really are and why are [clears throat] they called squares.

[00:04:19]So now that we know that which lockers remains open, let's understand what exactly are square numbers. A square number is formed when we multiply a number by itself. A number multiplied by itself. What does that mean?

[00:04:39]Thinking of it like a square shape.

[00:04:41]Let's think about a square shape. If each side is three units, if each side is three units, 1 2 3 1 2 3 1 2 3 and 1 2 3. You form a square. Then the total small squares will be 3. Three and three. How much it will be? There will be nine squares. And total small squares will be 3 into 3 that is 9 squared.

[00:05:07]Three from each side we have taken. And how do we write it? Simply we write it like 3 squared or simply 3 to the power two. The power two means the square or simply we say there are nine squares.

[00:05:23]And if what if we take the four squares like 1 2 3 4 sorry I'll take four from all the sides from here. 1 2 3 4 1 2 3 4 1 2 3 4 and 1 2 3 4 How many squares do we have in total from here till here? We have 16 squares. We have 16 square that what does that mean? 4 square is 16.

[00:05:51]That means the four square is 16.

[00:05:53]Similarly 5 square is 25. Now you tell me what will be 6 square.

[00:06:00]Yes it will be 36. What will be 7 squared?

[00:06:04]It will be 49. So in general for any number n we write n into n as n² which is also read as n²ared like 6 into 6 will be read as 6 squared. 7 into 7 will be read as 7 squared. that in area in the square units are 3x 5 squared 3x 5 into 3x 5 or we take 2.5 squared which is 2.5 into 2.5 ultimately gives us 6.25.

[00:06:34]So basically the squares of the natural numbers are called perfect squares which are 1 4 9 16 25 are all perfect squares. Here is the table given 1 square is 1 2 square is 4 9 square 3 square is 9 and 16 and 25. Let's quickly complete the table. 6 square 36 7 square 49. Say it with me. 8 square 64 9 squared 81 10 squared 100 11 squared 121. You have to repeat with me. 12 squared 144 13 squared 169 14 squared 196 15 squared 25 and so on.

[00:07:18]Now you tell me let's look at the last digit of these numbers. The last digit of these numbers on's place I mean on's place is 1 4 9 6 5 6 is again then 9 then four and then 1 and then zero again 1 4 9 6 and 5.

[00:07:40]You see they end with either 0 1 4 5 6 or 9. They're not ending with any other number. So a what does that mean? Other all these numbers ending with 0 1 4 5 6 or 9 are perfect squares.

[00:08:05]And the numbers which are not ending with 0 1 4 5 6 9 they are not at all a perfect square. Then how do we identify them? They end with 2 3 7 or 8. If a number ends in 0 1 4 5 6 9 is it always a square number?

[00:08:24]Tell me. The number 16 and 36 are both squares with six in the unit place.

[00:08:29]However 26 whose unit digit is six also not a square. Therefore, we cannot determine if the number is a square by looking at the digit in the on's place.

[00:08:39]But [snorts] the on's place can tell us whether a number is not in a square. If a number ends with 2 3 7 or 8, then we can definitely say that it is not a square. But if the number ends in 0 1 4 5 6 or 9, we can still take a chance to clarify whether the it ends with a whether it's a perfect square or not.

[00:08:58]Write five numbers such that you can determine by looking at the digits that they are not perfect square that they are not perfect square. So any number you write that should end with 2 3 7 or 8 like if I write 12 directly I can say it's not a perfect square because it's ending with two. If I write 13 or if I write 77 or if I write 98 by just by looking at the on's place of these I can definitely say they are not perfect square.

[00:09:27]I hope I'm clear.

[00:09:30]The squares 1 square 9 square 11 square 19 square 21 square 29 square all have one in their units place.

[00:09:39]Write the next two squares. Notice that if a number has one or nine in the unit's place, its square ends with one.

[00:09:47]If a number has ending with six or four, its on's place will be six.

[00:09:54]Let's understand this more now. Let's understand it more clearly. Now if we have a number with on's place like 1 or 9 I'll take those number like 1 square or 9 square or something like 99 square or something like 81 square or something like 69 squared where the ones place either 1 or 9 1 or 9 when we will calculate its square like 99 into 99 or somewhere 81 into 81 what will be the on's place here what will be the on's place the on's place will be one by because 9 9 are 81 the on's place will be one or if we have 81 into 81 we will have again the on's place will be one we are only concerned about the on's place here what the whole number will be we don't have to find out we just concerned about the on's place of a number which will be squared and which the answer will come what will be the number at the on's place after it it is being squared suppose if we have numbers ending with 2 or 8 like 32 squared or we can take 78 square or we can take 62 square. The number ending with 2 square 8 square 2 square what will come at its on's place?

[00:11:08]Which number will come? If you think of two square it is four. If you think of 8 square that is 64. In both the places what number is coming? Four. So any number ending with 2 or 8 will have four at its on's place. Similarly, if any number is ending at 3 or 7 like 63 squared or 77 square or 13 squared or 27 square, what will come at its on's place? That is 9. The number ending with 4 or 6. If you think 4 square that is 16. If you think of 6 square that is 36.

[00:11:43]What is common here? 6. So any number if you take 24 squared or 36 squared or 70 4 square any number ending with 4 6 or 4 and if you calculate it square the square number will also end with six and last but not the least five squared is 25.

[00:12:06]5 square is 25. So any number ending with five will have its square again ending with five only. No other choice.

[00:12:14]You have to remember all this. We don't have to calculate the whole square and then find out its one place. If we know the on's place, we can directly calculate what will be the ones place if the number will be squared.

[00:12:26]Now, which of the following numbers will have the digit six in the unit's place?

[00:12:32]For six in the unit's place, we just saw that we should have four or six at the on's place. So, we have 38. We have 34 46 56 74 82. Out of this tell me which numbers will have six digits at the units place?

[00:12:53]34 squared because 4 square is 16. 46 squared because 6 square is 36. 56 square because again 6 square is 36. And 74 squared which will have 44 are 16.

[00:13:08]But but why not 82 squared? Because if you see 82 square, it will have four at its on's place. If you see 38 square, it will again have four at its on's place.

[00:13:19]So we don't need this. We only need the six as its units place. We can find more by taking such numbers by ourselves. Now let's see one more property about the number of zeros. If we have if we take any number squared it has one zero like 20 squared or 30 or 40 or 50 or 7 10 when the square will be calculated we don't need the answer because we are talking about the number of zeros when the square will be calculated it will have double of zeros suppose if I have 7 1 0 squared if I have 7 1 0 square I don't want the answer I only want how many zeros will it have? It will have two zeros. But what if we have two zeros in the square number like 200 squared or 7100 squared or 100 squared for that matter? It will have four zeros. And if a number contains three zeros at the end, how many zeros will it square will have? 3 and 3 that is six zeros. What if we have four zeros in the end? it will have eight zeros. So what do you notice about the number of zeros at the end of the number and the number of zero at the end of its square? Will this always happen?

[00:14:39]Yes, this will always happen. Can we say that the square can have only even number of zeros? Yes. When you will calculate the squares, you either you will have two zeros or four zeros or six zeros or 8 or 10 and so on. So what can you say about the parity of a number and its square? If a number has any number of zeros, the squares will get in the square number, the number of zeros gets doubled. The number of squares gets doubled. I hope this is clear. Now let us explore the difference between the consecutive squares. What do you notice?

[00:15:16]We have two square numbers. 4 - 1 that is three.

[00:15:21]Now what do we see that if we subtract any two perfect square number we get an odd number like 3 5 3 7 or 9 and we're subtracting the perfect square number from this what we observe that when we add any consecutive odd number starting from one we get a consecutive square number. What does that mean? If we take one odd number the answer is one only.

[00:15:44]It's a perfect square. If we take two first odd numbers, we get the sum as four, which is 2 into 2. If we take the first three odd numbers, the sum will be 3 into 3. Without adding, I'm telling you, if we have taken the first four odd numbers, the sum will be 4 into 4. If we take the first five odd numbers, the sum will be 5 into 5. If we take the first six odd numbers, the sum will be 6 into 6. What if we take the first 10 odd numbers?

[00:16:12]sum write in my comment box what will you get this question is for you if you take the first 10 odd numbers and take their sum without calculating the sum what the answer will be I'm waiting now do you remember the pattern from grade six you have seen this picture in grade six ganitarash also where the pattern is when we add 1 and the next odd number three 1 + 3 then the next odd number is 1 + 3 and then Five. Here it is. Then we add 1, then three, and then five, and then seven. What does this show? This show that we are adding the first n odd numbers, which gives us the sum as n². What does that mean? If we add the first three odd numbers, we get the square of three, which is 9. Without adding, I'm telling you, if we add the first four odd numbers, 1 + 3 + 5 + 7, we get the sum of uh square of four, which is 16. So without adding we can say that the square of a number is the sum of the successive odd numbers starting from one. We can alternatively find out that whether a number is a perfect square by successively subtracting odd numbers. Now what does that mean? This means square finding square as a method of repeated subtraction. If I take any number suppose if I take 16 and I keep subtracting the odd numbers starting from 1. So let us start 16 - 1 I get 15.

[00:17:42]Next odd number subtraction that is 3 which I get 12. Next odd number that is five I get seven. Next odd number I take seven I get zero. So in the end if I get zero that means from where I started 16 is a perfect square number. But what if I don't get zero? If I don't get zero that number is not a perfect square. In your book the example has been given for 25. We all know that 25 is a perfect square. So here the example is that 25 we subtracted one we get 24. Then we subtracted the next odd number three we get 21. Next we subtracted the next odd number five we get 16. Next we subtracted the next odd number seven we get 9. And the next odd number we subtracted we get zero. So now we can see that 25 is the sum of 1 3 5 7 9.

[00:18:30]Five odd number. And we hence we can say that it's a perfect square. So we subtracted the first five odd number that's why 25 is the square of five. I hope this is clear. We we call it as a finding the square with the help of repeated subtraction. Finding the square root. Now let's see something else. Now using this pattern we can find 36 square but we have been given 35 square.

[00:18:57][snorts] From this question we know that 1 225 is the sum of first 35 odd numbers. That means if we take the first 35 odd numbers we get the sum of 35 squared. That means for 36 square we need the 36th odd number also. So how do we get to know what is the 36th odd number? How do we find directly? Instead of counting 1 3 5 7 9 11 keep going on and on. How do we find that? What will be the 36th odd number? So let's see how do we find that. The first odd number is 1. The second odd number is three. The third odd number is five. So what will be the nth odd number? The nth [snorts] odd number will be 2 n minus one. Now if we have to find the 36th odd number that means we will multiply 2 into 36 - 1 which will give us 71.

[00:19:48]So consider number. Now if we take the 38 is not a square number and we keep on subtracting odd numbers starting from one. If you see we subtracted 1 3 5 7 9 11. Now we got the negative number but in between positive and negative we did not get zero. That is why 38 cannot be expressed as the sum of consecutive odd numbers starting from 1. So from this page we got two things that perfect squares are the sum of odd numbers starting from one consecutive odd numbers and the second thing [snorts] that to find the nth odd number we have the formula 2 n minus one. These are the two things main things of this page that a perfect square by successively subtracting odd numbers and the nth odd number that is 2 n minus one and the numbers which are not perfect square by subtracting continuously repeatedly the odd numbers we will still not get zero.

[00:20:50]I hope this is clear. Let's move on to next. So now we can say that a natural number is not a perfect square if it cannot be expressed as the sum of successive or natural numbers.

[00:21:01]Successive or natural numbers starting from one like consecutive natural number starting from one. And we will use these results to find whether a natural number is a perfect square or not. This method is also known as repeated subtraction method.

[00:21:17]Repeated subtraction method. [snorts] Okay. Moving forward. Find how many natural how many numbers lie between two consecutive perfect squares. Do [snorts] you notice a pattern for this? I have taken this table from our previous videos previous of the same chapter where we have taken the squares 1 square 1 2 4 3 square 9 4 square 16 25 36 49 64 81 and 100. One row only I'll do and they tell you how it is to be calculated. If you want to know [snorts] how many numbers lie between two consecutive squares like 1 and four there are there's only two and three that means only two numbers lie between 4 and 9 5 6 7 8 that is only four numbers lie between these two suppose if I ask about 16 and 25 17 18 19 20 21 22 23 24 that's it that means eight numbers lie so without even calculating these 36 49 64 81 how many numbers lie between 81 and 100 it would be 18 excluding 81 excluding 100 in between numbers will be 18. But how about this line where I have not calculated any of the squares but still I want to know how many numbers will lie between any two consecutive squares like this or maybe this or maybe this. [snorts] So if you understand the pattern here what we are doing between any two square numbers the f how many numbers are lying the double of the first number like 2 into 1 two numbers are lying or between 2 square and 3 square the double of 2 that is four between four square and 5 square the double of four that is 8 between 6 square and 7 square the double of 6 that means the 12 numbers lie between 6 square and 7 square without even calculating. I don't want to know what 6 square is, what 7 square is and then I'll count on my fingers or maybe subtraction. I don't want to do that. I have a formula for this that between n² means any number and the next number n + 1 square the two consecutive squares how many numbers lie between those. double of the first number. For [snorts] just use this method between 12 square and 13 square the double of the first number that is 24 numbers lie between these between 14 square and 15 square the double of the first number that is 28 numbers lie between these two I'm not calculating what is 14 square what is 15 square and then subtracting or maybe counting on the fingers I'm just using the double of the first number like 17 square and 18 square it will be double of 17 that is 34 or may be between 21 square and 22 square the double of 21 that is 42.

[00:24:02]So this is one of the property very important just write down in your book itself or maybe your notebook which you are carrying just to make your notes.

[00:24:13]I hope this is understood. Double of the first number to calculate the numbers between two consecutive perfect squares.

[00:24:25]Now the next thing you will be doing yourself that how many square numbers like 1 4 9 16 25 36 these are called square numbers are there between 1 and 100 10 100 1 to 200 200 1 to 300.

[00:24:37]Complete the table from 1 square to 30 square or maybe 31 square till 32 square to find out how many numbers are there between 1 and 100. How many square numbers? Calculate all the squares first. Complete the table. till here 30 square and do one more 31 square and 32 square as well and then find out how many square numbers are there like 1 to 100 may 1 4 9 16 25 36 49 64 81 100 all these are from 1 to 100 then 100 1 to 200 it will be from 11 square to 14 square from 15 square 225 will start and it will go on till 17 square so find out yourself and fill the table.

[00:25:24]Now this is the other half of the previous page. I have taken in the two slides. Now perfect squares and triangular numbers. This triangular numbers also we have done in class 6.

[00:25:33]What are triangular numbers? These are the triangular numbers. How these numbers are formed? We take 1. Then we add the next counting number 1 + 2 which becomes 3. Then we add the next counting number 1 + 2 + 3 which becomes 6. Then we take the four counting numbers 1 + 2 + 3 + 6 that is 10. So these are the counting numbers. When we add triangular numbers when we add the counting numbers and what is the relation between triangular numbers and square numbers how do you form? If you see if you have taken the first two triangular numbers you get four. 1 and three makes four. The series is here. 1 and three will give you four the perfect square number. 3 and six will give you 9. The next perfect square number 6 and 10 will give you 16. The perfect square number 10 and 15 will give you 25. The next perfect square number. So extend the pattern show and draw the next term. Here you can see 1 and 3 total four. Here you see three.

[00:26:37]Here we have three and taken the six together that is 9. 3 and 6. 9. 6. Then we have six. 1 2 3 3 to 5 and 1 6 6 and 10 16. So draw the next pattern and find out yourself to find the 25 which is 5 square.

[00:26:56][snorts] Let's start with the major topic of this chapter. Now we have done square. Now we will do the square roots.

[00:27:03]What is a square root? Let's understand.

[00:27:06]Now let us understand what a square root is. A square root tells us which number was multiplied by itself to get the given number. It is the number which when multiplied by itself gives the original number. We can say that square goes forward square root comes backward.

[00:27:23]We all know that 7 into 7 is 49. So for getting that 49 perfect square number which number was multiplied by itself? 7 was multiplied by itself to get that number of 49. So we can say that square root of 49 is 7. Which number was multiplied by itself to get 25?

[00:27:45]We will denote it like this. This will be the root from which number this number has come when it was multiplied by itself. So that number is five. Can anyone guess which number was multiplied by itself to get that 100? That is 10.

[00:28:00]So this is a square root. Let's move forward. So the length of a side of a square with an area of 49 will be 7 cm because the area of square is side multiplied by side. So 7 was multiplied with 7 to get an area of 49. So we call the seven as the square root of 49 and x is the if y is the square of x then x is the square root of i as we say moving backward what is the square root of 64?

[00:28:29]We know that 8 into 8 is 64. So 8 will be the square root right? But what about - 8 into - 8? That is also 64. So we can say that 8 square is also 64. - 8 square is also 64. So the square roots of 64 are now 2 + 8 and minus 8. Every perfect square has two integer square roots. One is positive and the other is negative.

[00:28:54]The square root of a number is denoted by this radical sign as I discussed with. So now root 64 will be plus - 8.

[00:29:03]100 will be + - 10. Roo<unk> 8 square we can say is + - 8 or we can say roo<unk> 10 square is + - 10. In general for any squared number the square root will be plus and minus because the minus square also becomes a plus. But for this chapter we will only consider the positive square root and senior classes you will use the negative side of it.

[00:29:26]But for now this will be the the square root will only be considered positive not the negative.

[00:29:33]Now how do we get to know that if a number is a perfect square or not? How we have seen a property if the number ends with 2 3 7 8 that will definitely not be a perfect square. But if doesn't ends with 2 3 7 8 then there must be some method to see that whether it satisfies the condition of a square or not. So how to go about that? Let's see.

[00:29:57]Given a number such as 576 or 327, how do we find out that if it's a perfect square? If it is a perfect square, how do we get to know its root? We know that perfect squares end in 1 4 9 6 5 or even an even number of zeros also. We have studied earlier. But it is not certain that a number that satisfy this condition is a square. It is not certain. We can predict that it might be a square but that does not give the 100% sure. And we can clearly say that 237 is not a perfect square because we cannot say but we cannot be sure that 576 is a perfect square.

[00:30:34]Now we can list all the square numbers in sequence and find out whether 5 676 occurs amongst them. We know that 20 square is 400. We can find square of 21 22 23 and so on until we get 576 or a number greater than 576. This process becomes inefficient for larger numbers.

[00:30:52]Suppose if you get a four-digit or a five-digit number, how to go about it?

[00:30:55]The second method we studied that how repeated subtraction we can keep on subtracting the odd numbers starting from one. But at somehow this will also be very time consuming. Then we have to come to one method which will be giving us the correct square root in lesser time.

[00:31:17]Now if we know that the perfect square is obtained by multiplying an integer by itself. So what if we look at a number's prime factorization? Will it help in determining whether it's a perfect square? Yes, we can divide the prime factors of a number into two equal groups. The product of prime factors in either group combined to form the square root. What does that mean? Suppose if we take a number 324 and we calculate its prime factorization, we get prime factorization as 2 2 3 3.

[00:31:46]Then we will make groups out of it. Like we can do the group of two, a group of three and a group of three. From each group one number will come out two and then multiply it by three and then one number three will also come out. When we multiply 2 3's are 6 3's are 18. We see that 18 is the square root of 324.

[00:32:11]In this book they have made two groups of 2 into 3 into 3. 2 into 3 into 3. I have taken a group of two, a group of three and a group of three. And from each group I have taken one number to calculate its root. See what I did it's done here as well. So 324 is the square square root of 324 is 18. Now if we have to check 156 is a perfect square or not.

[00:32:34]We take the prime factorization as 2 2 3 and 13. We can make a group of two but we cannot pair three. We cannot pair 13.

[00:32:42]So we say that 156 is not a perfect square. Similarly, if we see 100 56 2800 whether they are perfect squares using prime factorization or not, we can estimate the square root of the largest perfect square by looking at the closest perfect squares we are familiar with and then narrowing down to the interval to search. For example, what to find root 936, we can reason as follows. 1 936 is between 1,600 and 2500. that is between the square root of 40 uh between 40 and 50 the square root of 1 936 will lie and the last digit of 1 1936 is 6. So last digit of the square root will be either 4 or 6. So it can be either 44 or 46.

[00:33:26]And if we calculate 45 squared we can compare it with 936 to half the interval to search from 40 to 50 to [snorts] either 40 to 55 or 45 to 50. 45 squared is written as a square + b square + 2 a b which is ultimately giving us 2 and 20 25. Now 2025 is greater. So 40 1 936 square root will lie between 40 and 45.

[00:33:51]From this observation we can guess and then verify that 1 936 is actually 44.

[00:33:56]It is more than 40 square and lesser than 45 square. That means it will not be 46 square it will be 44 square.

[00:34:05]Let's see one situation. Aribam and Bjo play a game. One says the number, other replies with its square roots. Aribam says starts he says 25. Bij quickly reply responds with five. Then Bij says 81. Aribam answers 9. The game is going on. The game goes on till Aribam says 250. Bij is not able to answer because 250 is not a perfect square. Arubam asked Vij if he can at least provide a number that is close to the square root of 250 then BO needs to estimate the square root of 250. We know that 250 lies between 100 and 400. 100 square root is 10 and 400 square root is 20. So that means 250 square root lies between 10 and 20. We still not very close to the number whose square is 250. We know that 15 square is 225 and 16 square is 256. Therefore 15 is lesser than square root of 250 lesser than 16. Since 256 is much closer to 250 than 225 then approximately 16 we know that it is less than 16 but at least it is approximately closer to 16.

[00:35:14]Another problem Akil has a square piece of cloth of area 1. He wants to know if he can cut a square handkerchief of side 15. If not he wants to know the maximum size of handkerchief that he can be cut from this piece of cloth with an integer side length and we all know that 125 is not a perfect square. The nearest perfect square will be R squares are 11 square 121 and 12 square is 144. So the largest square handkerchief with integer side length that can be cut will be one of side 11 cuz 11 square will be 121 170 121 cm square. From each side they can cut 11 cm of length and then they can get a largest hanker chief. Then we start with an exercise here.

[00:36:01]Which of the following numbers are not perfect squares? Remember the numbers ending with 2, 3, 7 and 8 are not perfect squares at all. So this is ending with two. So not a perfect square. This is ending with eight. Not a perfect square. This is ending with seven. Again, not a perfect square. This is ending with nine. So we can check this can be a perfect square or cannot be a perfect square. depends on how we calculate the prime factorization and its pairs.

[00:36:35]Which among one among 64 square, 108 square, 292 square, 36 square has last digit four. And I told you which number squares will have last digit four. The numbers ending with two square the number ending with 8 square. Any number ending with two ending with eight it square will have four.

[00:36:55]So 108 squares and 292 square will have the last digit four. And what about 64 square? 64 square will have the last digit as six. Same with 36 square will have the last digit as six. So we don't need that. Similarly, you can try all those questions. Find the length of the side of the square whose area is this.

[00:37:15]That means for side you have to calculate the square root of 441. For that you can use the prime factorization of method. Find the smallest square number by which that is divisible by each of the numbers 4 9 and 10. So we will calculate the LCM of 4 9 and 10.

[00:37:33]We'll see the prime factorization and then complete the square parts to calculate the smallest square number.

[00:37:40]How many numbers lie between the squares of the following numbers? We have done this property. Between 16 square and 17 square, 2 into 16 numbers will lie.

[00:37:49]between 99 square and 100 square into 99 numbers will lie. So that's how you'll be completing the exercise. For any doubt you can comment in the box and I'll give you the solution of it. In the following pattern fill in the missing numbers. If you see the pattern so here we see that we have a pattern 1 square + 2 square + 2 square is equal to 3 square. Observe 2 square + 3 square it leads to 6 square + and then 7 square.

[00:38:19]Mark my arrows 3 square + 4 square + 12 square then next is 13 square. If you observe the first two multiplication gives the third addition and then the consecutive number of it like 2 square + 3 square 2 3's are 6 square and then 6 square next number 7 square 3 4 are 12 square 12 square next number is 13 square. Now from this you can guess which number is going to come next. 4 square + 5 square 4 5 are 20 square and then 21 square 9 square + 10 square 9 10 are 90 square 90 next is 91 square. So that's how you will be doing it. You have to decode the pattern and then write your answer. Now there are five * 5 that is 25 tiny small squares here 1 2 3 4 5 6 7 8 9 in one line there are 9 into 25 tiny squares tiny squares I'm talking about and how many lines are there 1 2 3 4 5 6 7 8 9 so for all the tiny squares we have 9 into 25 nine times what do we have to count how many tiny squares are there in the following picture write the prime factorization So for 9 into 9 into 25 we can say 9 as 3 into 3 we can say another 9 as 3 into 3 we can say 25 as 5 into 5. This is the prime factorization and then you can multiply and find out how many tiny squares are there. With this we end our topic of squares and square roots. Square roots are the opposites of squares and we calculate the square roots by prime factorization method. And now we will start with the next topic cubic numbers.

[00:40:08]Now let's understand another interesting concept cubes and perfect cubes. Cube means multiplying a number by itself three times. That means if you multiply any number like three if I'm taking if I multiply number any number three times I get a cube of three which means 3 into 3 into 3 will be 27. If I multiply 2 into 2 into 2 three times I get a cube of two that is called 8. So 8 is called the cube of 2, 27 is called the cube of three. That's how. So square means multiplying a number two times and cube means multiplying a number three times.

[00:40:47]And what are perfect cube? We have heard of perfect squares. The numbers which we obtain by multiplying any given number twice, we get a perfect square. So the numbers 1 8 27. How do we get that? 1 is 1 into 1 into 1. 8 is 2 into 2 into 2. 3 27 is 3 into 3 into 3. So [snorts] these are the numbers which we get after multiplying a given number thrice. They are called perfect cubes.

[00:41:15]I hope this is clear. The number for found by multiplying a number itself three times are called perfect cubes.

[00:41:20][snorts] Not every number is a perfect cube. Why?

[00:41:24]Why 8 is a perfect cube? 27 is a perfect cube. 9 is not a perfect cube. Can anyone tell 9 is not a perfect cube? Is 9 a cube? We see that 2 into 2 into 2 is 8. 3 into 3 is 27. This shows that 9 is not a perfect cube. Nor is any number from 10 to 26. [snorts] After 8 we get the straight away the next number perfect cube as 27. That means from 9 to 26 there is no number which we get by multiplying any given number thrice. So no number multiplied three times gives us 9. So we cannot we say that 9 is also not a perfect cube. I hope this is clear.

[00:42:05]Now let's see further how many unit cubes are there in this unit cube of three units. What does that mean? In the first layer we have three three and three cubes that are nine cubes. We have from here to here we have nine cubes and then again one more layer of nine cubes and then again one more layer of nine cubes. So that means we have arranged nine cubes three three cubes three cubes and three cubes that means in total we have arranged 3 into 3 into 3 that means 27 cubes and this one has 64 cubes. How? If you notice carefully each layer of this cube has 4 into four unit cubes like four cubes here, four cubes here and this one layer has from here till here this layer has 16 cubes. Each layer of this cube has four into four unit cubes. Each square layer has 16 unit cube. One layer of 16.

[00:43:04]Then another layer of 16. Then another layer of 16. Then another layer of 16.

[00:43:09]That means four layers of 16. That means 4 into 4 are four times that is 64.

[00:43:15]Similarly 5 cube is 5 into 5 into 5. 125 is a cube. In general we say for any number we write the cube as n * n * n as n cube. We put a three [snorts] supererscript that means n cube. Let's complete the table. 1 cube is 8. 2 cube is 27. 5 3 cube is 23. 3 cube is 27. 4 cube is 64. 5 cube is 125. 6 cube is 216. 7 cube is 3 43.

[00:43:43]8 cube is 512. 9 cube is 729.

[00:43:48]10 cube is 1,000. Now the next row will be completed by you. Do drop a comment if you don't want to know any of the cube. What patterns do you notice in the table? We know that 0 1 4 5 6 9 are the only last digits possible for squares.

[00:44:06]But what are the possible digits last digits of cubes?

[00:44:11]Tell me what are the possible last digits of the cubes? If you see the previous page.

[00:44:16]So now we already know that the square numbers can end only with certain digits. My question is what are the possible last digits of cubes? We all know that 1 cube is 1, 8, 2 cube is 8, 3 cube is 27, 4 cube is 64, 5 cube is 125.

[00:44:33]Do you notice the last digits? 1 8 7 4 5. If you continue further, you'll observe something amazing. Cubes can actually end with any digit from 0 to 9.

[00:44:45]Now let's think deeper. How many cube numbers will have one digit?

[00:44:50]Only these two cubes will have one digit. One and two cubes. How many cubes will have twodigit numbers? Only these two, three cube and four cube. Again only two. Now the three-digit cubes like 5 cube is 125 till from 5 cube till 9 cube all the numbers perfect cubes will be three digits. So there are five three-digit cubes. And what will be the next cube after 729? Just wrote it under the at the previous page. Yes, 10 cube which is 1,000 a first fourdigit perfect cube. Now the next thing is can a cube end with exactly two zeros. This question I have just explained to you verbally can similar to squares can you find the number of cubes with one digit two two digits two three digits five.

[00:45:39]So now the next is can a cube ends with exactly two zeros. Now this is a very interesting question like 100 900 if you see these numbers can it end with like this 9500 or 900 something like with two zeros tell me to get zeros at the end the number must contain the factors of 10 like 10 is 2 into 5 in cubes factor comes in the groups of three like in square the factors come in the groups of two in cubes the factor comes in the groups of three so zeros in cubes usually comes in the multiple of three like 10 cube which is 1,000 and 100 cube that is 1 lakh 10 lakh it's six zeros. So cube cannot end exactly with two zeros. It will end with either three zeros or six zeros.

[00:46:30][snorts] Now cubes of fractions decimals these are the fractions these are the decimals these are the negative numbers.

[00:46:38]So just like whole numbers fraction and decimals are can also be cubed. Like fraction if you give a fraction example we have 4x 6 cq. So that means we are multiplying 4x 6 into 4x 6 into 4x 6 three times. So numerator will be multiplied thrice that is 64.

[00:46:53]Denominator will be multiplied thrice that is 216. So that's how it's a very simple numerator multiplied three times.

[00:47:00]Denominator multiplied three times. Can you see the decimal example? Guess decimal number will be multiplied three times and we'll put the decimal as per the decimal places in the three numbers like we have two decimal places two decimal places and two decimal places.

[00:47:14]So in all after the six decimal places we'll put the decimal a negative number cube minus 6 cube that means if you multiply - 6 thrice integer multiplication anything any negative number multiplied odd number of times will give a negative number only. So when a negative is cubed the answer remains negative.

[00:47:35]Coming on to a taxi cap numbers a most interesting story. This story is about the great mathematician Shinasa Ramanujan.

[00:47:45]One day Ramanujan was ill in a hospital.

[00:47:48]His friend GH Hardy came to visit him in a taxi number 1729. He came in a taxi number 1729.

[00:47:58]Hardly Kardi casually said 1729 seems like a dull number but Rammanujan immediately replied no it's a very interesting number can anyone explain why why did he say that similar uh interesting number because 1729 can be written as 1 cube in addition to 12 cq.

[00:48:20]How smart he was and other way 9 cq + 10 cube. Very nice. So 1729 can be written as the sum of two cubes in two different ways and this makes it special. So because of this story 1729 has Since So 1729 has since been known as the Hardi Ramanojan number and that number can be expressed as the sum of two cubes in two different ways and are called taxi cab numbers. The next two cap taxi cab numbers after 1729 are 4104 and 13832.

[00:50:22]Now you have to find the next two ways in which each of these can be expressed as the sum of two positive numbers. So 1729 is the smallest taxi cab number and today it is will be called Hardy Ramanujan numbers.

[00:50:40]Now let's discover a very beautiful pattern hidden inside these cube numbers. We have already learned that the square numbers are connected with odd numbers. But did you know cube numbers also follow a pattern with odd numbers? Let's look carefully. This 1 cube is 1. Then this 3 + 5 is 8 which is actually a 2 cube. This 7 + 9 + 11 is 27 which is actually 3 cube. The next four odd number sum is 64 which is 4 cube.

[00:51:09]Did you notice something amazing?

[00:51:12]For cube numbers we add consecutive odd numbers and the odd number added is equal to the cube root and not starting from one again. For 3 cube we add three odd numbers. For 5 cube we added we added five odd numbers.

[00:51:28]Can you guess this one? 31 + 35 33 35 37 it's given it will be 216 that is 6 cube. Now the book asked an interesting question without calculating can you tell the value of 91 to 109 all odd numbers cube sum. Let's observe carefully these are the consecutive odd numbers. How many numbers are there?

[00:51:54]Count them. From 91 we have 1 2 3 4 5 6 7 8 9 and 10. there are 10 odd numbers.

[00:52:01]So this sum will be 10 odd cubes that is 1,000. I hope this is clear. So without calculating the sum we have seen that it will be 10 cube that is 1,000.

[00:52:15]Firstly we took one odd number then we took two odd numbers then we took three odd number then we took four odd numbers five odd numbers five cube then we took six odd numbers 6 cube and without just taking out the sum we saw that we have taken 10 odd cube 10 odd odd numbers that will be 10 cubes which is 1,000.

[00:52:38]Coming on to the next topic which is cube roots.

[00:52:43]Let's understand what the cube roots in a very simple way. We know that 2 into 2 into 2 is 8. So 2 is called the cube root of 8 and it is denoted like this.

[00:52:54]Like this will be the square root. But inside if we write like 3 and cube root of 8 that is 2. 3 cube is 27. So we can say cube root of 27 will be 3. Cube root means which number will be multiplied by itself three times gives the original number. So that number is called the cube root. Now how do we know whether a number is a perfect cube or not? How do we find out? Again taking inspiration from the case of square. How do we used to do that? We used to take the prime factorization. So we will use it here as well. So [snorts] if we take the number 3 375 where we have to check whether it's a perfect cube. We have taken its per prime factorization. Now the factors have to be split in a group of three.

[00:53:36]Can we make a group of three? Here we have three three times. Five three times. Yes. So all the factors are arranged in the triplets. A triplet is here. A triplet is here. So for 3375 we can form three group from the groups of three into five. That means it will be a cube of 15.

[00:53:56]Is 500 a perfect cube? If we see the prime factoriation of 5 100 it is double triple five. We have a triplet of five but we don't have a triplet of two. We don't have a triplet of two. That means we cannot be split into three identical groups. That's why we cannot say 500. We can say that 500 is not a perfect cube.

[00:54:16]So today we have learned that the pattern of cubes with odd numbers, the cube roots and how to identify whether the number is a perfect cube or not.

[00:54:24]Just a memory trick I'm giving you.

[00:54:26]Square needs pairs. Cubes needs triplets. Now let's look at another important property of perfect cubes.

[00:54:35]Look carefully at this table. On the left side, we have the prime factors of a number, any number. And on the right side, we have the prime factors of its cube.

[00:54:46]For example, the four the prime factors of four are 2 into 2. And now when we cube the four, we get 2 into 2 into 2 into 2 that is 610. What we are doing, which can also be written as 2 cube into 2 cube. So each factor we have cubed. It has it had two factors. So we cubed we cubed. Similarly 6 the prime factorization is 2 into 3. When we did the cube of 6 that is 3 * 2 and 3 * 3 that is 2 cube and 3 cub directly we can say that we cubed two we cubed three.

[00:55:19]You notice something that each prime factor is appearing three time in it cube.

[00:55:24]So again each factor is again appearing in its triplets. If we see for 15 as well we see for 12 as well it's repeating. So prime factors always occurs in group of three. Again memory line perfect square comes in pairs.

[00:55:41]Perfect cube come in triplets. Now coming on to the next thing which is successive differences. What is successive differences? Let's understand it. Now we know perfect squares are 1 4 9 16 25. Let's take the difference between the two consecutive perfect squares that is 4 - 1 3 9 - 4 5 16 - 9 is 7 9 11 so on further let's subtract these numbers again 5 - 3 is 2 7 - 5 9 9 - 7 11 - 9 and after second level we get the constant two here after two levels the difference becomes the constant and the constant number is two but what if we do with the perfect cubes. Let's try the same with the perfect cubes. We have 1 8 27 64 125 and so on. If we take 8 - 1 that is 7 27 - 8 19 37 61. Now let's subtract again. 19 - 7 12 then 18 then 24 let's do one more step. 12 18 - 12 6 18 24 - 18 6. So third differences will be 6 66 6. So for cubes the difference becomes constant after three levels and the constant value is six. For squares the difference becomes constant after two levels and the difference is two.

[00:57:12]Squares are connected with two cubes are connected with three. That's why square becomes constant after two levels. Cubes becomes constant after three levels. A pinch of history let's discuss. Did you know that the perfect squares and cubes were studied thousands of years ago? The Babylonians made tables of squares and cubes around 1700 B.CE. And these tables had have helped people in land measurement, architecture, calculating roots quickly. So what we have learned today that the prime factors of the cubes, cube root, successive differences and a little bit of history of mathematics.

[00:57:49]Don't forget the pair of squares, the triplet for cubes, the two level of for squares and the three level for cubes.

[00:57:58]Now in ancient India some special Sanskrit words were used for these ideas which we just discussed like land measurement, architectural designs and everything. The Sanskrit word va. If you see the Sanskrit word va which was used for both the square shape and the square number.

[00:58:18]Square of a number and a square shape.

[00:58:20]Like if we say five square is 25. This is called vga. And the another word the four like if they take the fourth power it is called va. Va. And what is ghana?

[00:58:31]Ghana. The Ghana word was used for cube shapes or we can say cube numbers like 3 cube is 27. So we called used to call that as ghana. What was Arya's contribution? Let's see. One of the India's greatest mathematicians Aryata explained that a square figure and a product of equal quantities are both called VGA. This shows that an ancient mathematicians connected geometry and the numbers together.

[00:58:58]If you see a square figure of four equal sides and a number representing its area are called VGA. The product of two equal quantities is called VGA. V. We use the term for square power has its original in the graphical representation of a square figure. Now why is the word root used for mathematical operation square root or maybe cube root? Why do we use the word root in the mathematics like square root or the cube root? In ancient India the Sanskrit word mle meant the root of a plant.

[00:59:28]But it also meant the origin, the base, the beginning. So mathematicians use the word mle for roots in mathematics too.

[00:59:37]Like vermul which means square root. If you see it here, ghamul which means a cube root. Later similar words were used in Arabic and Latin languages too. So the idea of mathematical roots spreads across the world. Coming on to the exercise. Let's some solve questions.

[00:59:58]So find the cube roots of these two numbers. You can use prime factorization and then you can take the triplets of number and then multiply each factor to get the cube roots of 47,000 and 10,648.

[01:00:15]It will be 30. This will be 22. Do it and check it from here whether you're getting the same answer or not. What number should 1 3 2 3 be multiplied to make it a cube number. So when you do the prime factorization of 1 3 2 3 you will see that you are getting 3 * 3 and 2 * 7. So something is missing here to make it a perfect cube. We have a triplet of three but we have a pair of seven. For a perfect cube factors must come in triplets. Three already has a triplet. So we multiply by one more 7 so that the number becomes 9 to 61 and when it becomes 9 to 61 it will have one more 7 and the cube root will become 21. It will be a cube of 21.

[01:01:01]Let's see something true or false and see our reasoning as well. The cube of any odd number is even. If we take 3 cube that is 27 odd number cube even not at all. So it's a false example I've just given you. 3 cube is 27. There is no perfect cube that ends with 8. 2 cube only ends with 8. So that means it's again a false situation. The cube of twodigit number may be a three-digit number. Yes. 12 cube.

[01:01:33]5 cube. I mean 5 cube is 125.

[01:01:43]The cube of a two-digit number may have seven or more digits. Largest two-digit number is 99. And if we take its cube, it will be 97029.

[01:01:53]97029 only six digits. So, it cannot have seven or more digits. Again, a false situation. Cube numbers have an odd number of factors. Only perfect square have an odd number of factors. Perfect cubes may or may not have odd number of factors.

[01:02:11]Now you are told that 1331 is a cube of perfect cube. Can you guess without factoriation what it cube root is?

[01:02:18]Similarly guess the cube roots of the other numbers. We know that 11 cube root is 1331. So we can say that 11 is the cube root of this number. 17 cube will be 4 913. 23 cube will be 1 to 1 167 and 32 cube will be 32768.

[01:02:37]So from here we have explored the history behind the square roots and the cube roots. The ancient in mathematics cube root questions and tricks ending I'll say that mathematics is not just a numbers it's also a history a language patterns and human creativity.

[01:02:53]Now let's solve one final thinking question from this chapter. Which of these is the greatest? We have four parts 67 cq - 66 cq and then the following parts. Let's notice something important. In every case, the numbers are consecutive like 66 67 43 42 67 66 square 43 square - 42 square. Let's use a shortcut method for these two parts for squares. 67 square - 66² a square - b square property if we use it we get 67 + 66 - 66. So this will be 133 into 1 which is 133.

[01:03:33]Same prop identity we apply for 43 square - 42 square it will be 85 into 1 1* addition 1* subtraction which will be 85.

[01:03:44]So among the square expressions 133 is greater that means this part is greater.

[01:03:48]Let's understand a and b part. Now look at the cubes. When numbers become largest cubes grow very fast. com let's compare roughly 67 cq minus 66 cq. Since 67 and 66 are much bigger, the difference cube difference will also be much bigger. The greatest so the greatest expression will be the 67 cube minus 66 cube. For consecutive numbers, square differences increase steadily.

[01:04:13]But for cube differences increases much faster because cubes grow faster than the squares. You have to keep this point in mind. So let's quickly revise the whole chapter. A square number is formed when a number is multiplied by itself.

[01:04:27]Perfect squares are the squares of natural numbers. All perfect squares ends with 0 1 5 4 5 6 or 9. Square root is the opposite of square. A cube number is formed when a number is multiplied by itself three times. A per perfect cube has a prime factors in triplets.

[01:04:50]Cube root means finding the original number.

[01:04:55]That's final memory. Square root goes backwards from square. Cube root goes backwards from cube. And with this we complete the chapter a square and a cube. If you have enjoyed learning with me, do like, share and subscribe for more mathematics videos. Remember mathematics is full of beautiful patterns. We just need to observe them carefully. Thank you.