Can You Solve this beautiful 'Harvard' Entrance Exam Question

Ajouté : 2026-05-26

[00:00:00]Hello everyone, you are welcome. Today we have a very beautiful simplification math question. What is the value of 60 raised to power 6 - 50 to power 6? So we'll try to simplify and solve this problem step by step and using standard exponential math properties. So let's start our solution. First of all here we can write these two powers as this will become 60 to power. 6 can be written as this is simply 3 * 2 minus 50 to power 6 is 3 * 2 and in both the terms you will use an exponential math property. So here we will use this one exponential map property. We can write a ra to power m * n as a ra to power m r to power n.

[00:00:59]So using this exponential math property here both the terms will become this will become 60 to power 3 whole² minus and this will become 50 cq whole squared.

[00:01:22]Look into both the terms here both the terms are in square form. So here in this stage we will use an algebraic identity a square minus b square. So I will try to use this one algebraic identity a² - b² that is a + b * a minus b.

[00:01:45]So using this algebraic identity here this expression will become this will become 60 Q + 50 Q* 60 cube minus 50 cube.

[00:02:07]Here we try to find out the value of 60 cube and 50 cube. Then we substitute that values here. So therefore here we can write 50 cq is what is the value of 5 cube? 5 cube is simply 5 * 5 is 25 and 25 * 5 is 125 and there is cube / 0. So we'll write this zero three times. So this is 125,000 and the value of 60 cq is here 6 cube is 6 * 6 is 36 and 36 36 * 6 is about 216 and we will write three zeros here. So this is 216,000.

[00:02:54]Now we substitute these two values here in this one expression. So this expression will become here the value of 60 cube is it is 216,000 plus the value of 50 cq is 125,000 no again the value of 60 cq is 216,000 minus 50 cq it is 125,000.

[00:03:26]Now let's add and subtract these two numbers. So it is very easy. There are three zeros in the on's period. So this will become three zeros and next is five. So 5 6 is 11. 1 will be carried to this digit. 2 + 1 is 3 + 1 is 4 and 1 2 is 3. So this is 341,000 times and these two numbers. So here 0 - 0 it is 3 0 and next is 6. So 6 - 5 it is 1 and here 1 - 2 is not possible. So 21 - 12 it is about 9. So this is 91,000.

[00:04:10]Now we'll try to multiply these two numbers without any calculator.

[00:04:14]So this is very easy and very interesting. So if I write this first number is the same 341,000 times here we can write this number as the place value of 9 which is 90,000 plus the place value of 1 which is 1,000.

[00:04:37]So here first we try to multiply this number with 90,000 and secondly with 1,000.

[00:04:44]So this is very easy here this number times 3 0 is that will be three zeros as the same next is this number * 0 is 0 this number * 9 is let's multiply 9 with this number so multiplying 9 with three zeros this will be 3 0 the same no 9 * 1 is 9 9 * 4 is 36 so 6 3 will be carried 9 * 3 is 27. 27 + 3 is 30.

[00:05:18]So this gives him 30 billion 690 million.

[00:05:24]Plus let's multiply this number with 1,000. So again it is very easy here this number times three zeros. This three zeros will be the same.

[00:05:35]Next is one. So multiplying one with this number. This gives him this number as the same.

[00:05:42]Next is 1 4 3. So this gives same 341 million. So finally we will try to add up these two numbers. That will be our final answer. So let's start from the once periods. There are three zeros and there are also three zeros. So there are three zeros will be the same. Next is again three zeros there and there as well. So again there will be three zeros. Next is 1. So 1 + 0 is 1. 4 + 9 is 13. There will be one carry. 3 + 1 is 4. 4 + 6 is 10. So this becomes zero.

[00:06:20]One will be carried to this one digit. 0 + 1 is 1 and this is 3. So this gives him 31 billion and 31 million. So finally the answer of this interesting simplification math question it is simply that is 31 billion and 31 million and that's our final answer.