Nice Simplification Math Problem | How to Simplify Step by Step?

[00:00:00]What is the value of whole square root of 4 raised to power 8 minus 2 raised to power 10?

[00:00:07]So, here we will try to simplify this problem step by step and without any calculator.

[00:00:12]So, let's start our solution. First of all, here we can write this 4 as this is simply 2 squared.

[00:00:19]So, this will become whole square root of 4 is 2 squared raised to power 8 minus 2 raised to power 10.

[00:00:31]And here we will use an exponential identity.

[00:00:34]So, this will become this is whole square root of 2 raised to power 2 * 8 this is 16 minus 2 raised to power 10.

[00:00:47]Next, we can write this 2 raised to power 16 as this is simply whole square root of this is 2 raised to power 10 * 2 raised to power 6 minus 2 raised to power 10.

[00:01:03]And there is 2 raised to power 10 common, so we will take out this common.

[00:01:08]So, this will become whole square root of 2 raised to power 10 times this is 2 raised to power 6 minus 1.

[00:01:19]So, here we know that we can write square root of a * b as square root of square root of a * square root of b for any two non-zero numbers a and b.

[00:01:34]So, this will become here this is square root of 2 raised to power 10 times square root of 2 raised to power 6 minus 1.

[00:01:46]Next, we can write this 2 raised to power 10 as this is square root of 2 raised to power 10 is same as 5 * 2.

[00:01:56]And here we What is the value of 2 raised to power 6?

[00:02:01]2 raised to power 6 This is simply 2 * 2 is 4 4 * 2 is 8 8 * 2 is 16 16 * 2 is 32 and 32 * 2 it is about 64.

[00:02:14]So therefore the value of 2 raised to power 6 it is 64.

[00:02:19]So this will become 64 1 And inside this one number we will use exponential math property. We know that we can write a raised to power m * n as a raised to power m whole raised to power n.

[00:02:39]So using this exponential property here this will become This will become whole square root of 2 raised to power 5 whole square times and this will become square root of 64 - 1 This is simply 63.

[00:02:55]But we can write 63 as this is 9 * 7. 9 * 7 is 63.

[00:03:01]So here we can cancel this square with square root so this will become only 2 raised to power 5.

[00:03:08]And here we can write this as square root of 9 * square root of 7.

[00:03:14]So what is the value of 2 raised to power 5? Multiplying 2 five times 2 * 2 is 4 4 * 2 is 8 8 * 2 is 16 and 16 * 2 is 32.

[00:03:24]So the value of 2 raised to power 5 this is 32. 2 So we'll substitute this value here.

[00:03:31]So the value of 2 raised to power 5 this is 32 * square root of 9 is 3 * square root of 7.

[00:03:41]And finally 3 * 2 is 6 3 * 3 is 9. So this is 96 * 7.

[00:03:50]So finally the final value of this simplification math question, it is simply This is 96 * √ 7.

[00:04:03]And that's our answer.