Solving a 'Harvard' University Entrance Exam Question

Indexed: 2026-05-21

[00:00:00]Hello everyone. You all welcome. Today we have a very interesting simplification math question.

[00:00:07]Here is what is the value of whole square root of 9 raised to power 9 minus 9 raised to power 8.

[00:00:15]So first of all here we can write this first term in the expression as 9 raised to power 9 but 9 can be written as this is simply 8 plus 1 minus 9 raised to power 8 whole square root.

[00:00:33]And in the first term we will use an exponential identity.

[00:00:37]So here in this step we will use this one exponential identity. We can write a raised to power m plus n as a raised to power m times a raised to power n.

[00:00:51]So using this exponential identity here this expression will become this will become 9 raised to power 8 times 9 raised to power 1 which is simply 9 minus 9 raised to power 8 whole square root.

[00:01:10]Looking at this expression here 9 raised to power 8 is common in both the terms inside the square root.

[00:01:16]So we'll take out this common. So this will become whole square root of taking 9 raised to power 8 common this become 9 minus and this will be only 1 left.

[00:01:29]So next we can write this as this is 9 raised to power 8 times here 9 minus 1 is simply 8.

[00:01:40]Here we will use a radical math property that is here we know that we can write whole square root of product of two numbers as the square root of the first number times square root of the second number.

[00:01:57]So using this radical property here, this will become this will become whole square root of 9 raised to power 8 * square root of 8.

[00:02:10]So, let's see here, we can write this power 8 and this 8 as this will become whole square root of 9 raised to power 8 can be written as this is simply 4 * 2 times and here this 8 is 4 * 2.

[00:02:29]So, further, here inside the square root, we will use this one explanation math property. We can write a raised to power m * n as a raised to power m whole raised to power n.

[00:02:45]So, using this explanation property, here this will become this will become whole square root of 9 raised to power 4 whole square times and using this radical property here, this will become square root of 4 * square root of 2.

[00:03:06]So, here we can cancel the square with square root.

[00:03:10]So, this will become 9 raised to power 4, but 9 raised to power 4 can be written as 9 squared * 9 squared times here square root of 4 is simply 2 and this will become square root of 2.

[00:03:26]And here 9 squared, it is simply it is simply 9 * 9, which is 80 1.

[00:03:35]So, we'll replace these two numbers with 81.

[00:03:38]So, this expression will become this is 81 times here we can write this next 81 as this is 80 + 1 times 2 * square root of two.

[00:03:56]And we'll try to multiply these two numbers.

[00:03:59]81 with these two numbers, so this is very easy here.

[00:04:03]81 times 80 is 81 times zero is zero.

[00:04:08]And 81 times eight is let's multiply eight with this. So eight time one is eight and eight time eight is 64.

[00:04:17]Plus let's multiply this with one. So here 81 times one is 81.

[00:04:23]times two times square root of two.

[00:04:28]So let's sum up these two numbers. So here it is very easy here. One zero is one.

[00:04:34]Eight eight is 16, so this is six.

[00:04:38]Four plus one is five and this is six.

[00:04:42]times two times square root of two.

[00:04:46]Let's multiply two with this one number.

[00:04:49]So we will start from the ones position here. Two time one is two. Two times six is 12. One will be carried.

[00:04:57]Two time five is 10. 10 plus one is 11.

[00:05:01]One will be carried. Two times six is 12. 12 plus one is 13.

[00:05:05]So this become 13,122 times square root of two.

[00:05:10]Which is our final answer.

[00:05:13]So finally the value of this expression and beautiful math question is This is simply This is 13,000 122 times square root of two.

[00:05:32]And it is our final answer.