Solving a 'Harvard' University Entrance Exam Question

Added: 2026-05-18

[00:00:00]Hello everyone, you are welcome.

[00:00:03]What is the value of this expression? 64 raised to power 3 minus 16 raised to power 4.

[00:00:10]So let's start our solution. Here we'll try to simplify this with something different methods.

[00:00:15]So first of all here we can write these two bases as here we can write 16 as This is same as 2 * 2 is 4. 4 * 2 is 8 and 8 * 2 is 16. So this is 2 raised to power 4.

[00:00:32]And we can write 64 as this is simply 2 * 2 is [clears throat] 4. 4 * 2 is 8. 8 * 2 is 16. 16 * 2 is 32 and 32 * 2 it is 64.

[00:00:45]So here this 2 is about six times.

[00:00:49]So here we'll replace these two bases with these two exponential numbers.

[00:00:55]So this expression will become here 64 is simply 2 raised to power 6 raised to the power 3 minus 16 is 2 raised to power 4 raised to power 4.

[00:01:09]And in both the terms we'll use an exponential identity.

[00:01:13]So here we'll use this one exponential identity a raised to power m raised to power n It can also be written as a raised to power m * n.

[00:01:24]So using this identity here this will become 2 raised to power 6 * 3 is 18.

[00:01:31]Minus this will become 2 raised to power 4 * 4 is 16.

[00:01:37]But then we have to simplify this expression. Here we can simplify this expression with different methods.

[00:01:43]One method is to find out the value of each term and then subtract both the terms. The second method is here we can take 2 raised to power 16 common from both the terms. But here we'll try to solve this with something different method.

[00:01:58]So, let's try this method. Here we can have this 18 this power is 2 raised to power.

[00:02:05]18 is same as 9 * 2.

[00:02:09]Minus 2 raised to power 16 is same as 8 * 2.

[00:02:16]So, applying this exponential math property here, we can write this expression as This will become 2 raised to power 9 whole square minus This become 2 raised to power 8 whole square.

[00:02:34]Looking to both the terms here, both the terms are in square form. So, therefore, here we'll apply this one algebraic math property.

[00:02:42]a² b² that is same as a + b * a - b.

[00:02:52]So, let's apply this algebraic identity here.

[00:02:55]So, here where a is 2 raised to power 9 and our b is 2 raised to power 8.

[00:03:00]So, this will become 2 raised to power 9 plus 2 raised to power 8.

[00:03:07]And further, this is 2 raised to power 9 minus 2 raised to power 8.

[00:03:14]Next, our target is to find out the value of 2 raised to power 8 and 2 raised to power 9.

[00:03:20]So, therefore, here we can write 2 raised to power 9 as This is same as 2 raised to power 2 raised to power 6 * 2 raised to power 3.

[00:03:34]Here we have the value of 2 raised to power 6 that is 64.

[00:03:38]So, we'll replace 2 raised to power 6 with 64 times and 2 raised to power 3 is simply 2 * 2 is 4, 4 * 2 is 8.

[00:03:47]So, let's multiply these two numbers.

[00:03:48]So, here 8 * 4, it 8 32.

[00:03:52]So, this is 2 and we'll be carrying. And 8 * 6 is 48. 48 + 3 is 51. So, this is 512.

[00:04:00]And we can write 2 raised to power 8 as 2 raised to power 8 is simply 2 raised to power 6 * 2 raised to power 2.

[00:04:10]2 raised to power 6 is 64.

[00:04:13]And 2 raised to power 2 is simply 2 * 2 is 4.

[00:04:16]And here 4 * 4 is 16.

[00:04:18]4 * 8 is 24. 24 + 1 is 25.

[00:04:23]So, this is 256.

[00:04:25]So, here these two expressions will become the value of 2 raised to power 9 is simply 512 plus this value is about 256 times 2 raised to power 9 is 512 minus 2 raised to power 8 is 256.

[00:04:49]So, let's try to simplify these two two expressions.

[00:04:53]So, let's sum up these two numbers. So, 6 + 2 is 8.

[00:04:58]5 + 1 is 6. And 2 + 5 is 7.

[00:05:03]times Here, this is 2 - 6 is not possible.

[00:05:06]This will become 12. 12 - 6 is 6.

[00:05:10]This will become 0. So, this will become 10. 10 - 5 is 5. This will become 4. 4 - 2 is 2.

[00:05:17]This is the same 256.

[00:05:20]So, let's try to multiply these two numbers. So, here we will write this first number as the same. So, 168 times and here we can write this number as the sum of 200 plus 50 plus 6.

[00:05:38]So, let's try to multiply this number first with 200. That is very easy and very interesting.

[00:05:44]This This times 0 is 0. This number times zero is zero. This number times two is two times eight is 16.

[00:05:53]This is six one will be carried. Two times six is 12. 12 plus one is 13.

[00:05:57]Two times seven is 14. 14 plus one is 15. So this gives him 153,600.

[00:06:04]Plus Let's multiply this number with 50. So this number times zero is zero. This number times five is five times eight is 40. Zero four will be carried. Five times six is 30. 30 plus four is 34. Four three will be carried. Four times seven is 35.

[00:06:21]35 plus three is 30. And this gives him 38,400.

[00:06:26]Plus Let's multiply this number with six.

[00:06:30]So here, let's start from the ones position. Six times eight is 48. This is eight. Four is carried. Six times six is 38. 38 plus four is simply 40.

[00:06:41]Zero four will be carried. Six times seven is 42. 42 plus four is 46.

[00:06:47]This will become 4,608.

[00:06:51]Finally, let's sum up these three numbers so that will be the correct and final answer.

[00:06:56]So here, eight zero zero is simply eight.

[00:07:00]Next is zero. So zero zero and zero is zero.

[00:07:05]And there is six. So six plus four is 10. Plus six is 16.

[00:07:09]This is six. One will be carried to this one digit. Four plus one is five. Five eight is 13.

[00:07:15]13 three is 16. This is six.

[00:07:19]And one will be come here. Three one is four.

[00:07:21]Four plus five is this is nine. And this is one.

[00:07:26]So this gives him 196,608.

[00:07:29]And that is the final value of this beautiful math problem.