Solve Nested Roots: Finding Fifth Root of Square Root of 1024

Video Solution

Solution Steps

00:06 Let's solve this math problem together.

00:09 A regular root is called a square root. It has an order of two.

00:14 If we have a number, A, under two roots: one of order B, and another of order C.

00:22 The answer is A, under a root of order B times C.

00:27 Let's use this rule for our exercise.

00:32 Now, if A is to the power of B, under a root order C.

00:43 The answer is A to the power of B divided by C.

00:47 We'll use this formula for our problem.

00:51 And there you have it. That's our solution!

Step-by-Step Solution

To solve this problem, we'll follow these steps:

Now, let's work through each step:

Step 1: Calculate the inner square root.
We have $\sqrt{1024}$ . We know that $1024 = 2^{10}$ , so $\sqrt{1024} = \sqrt{2^{10}}$ .

Applying the property of roots, $\sqrt{2^{10}} = 2^{\frac{10}{2}} = 2^5 = 32$ .

Step 2: Now, apply the fifth root to the result from step 1.
We need to find $\sqrt[5]{32}$ .

Step 3: Simplify using the properties of exponents.
From $\sqrt[5]{32} = \sqrt[5]{2^5}$ , we have $2^{\frac{5}{5}} = 2^1 = 2$ .

Therefore, the solution to the problem is $2$ .