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

Video Solution

Solution Steps

00:00 Solve the following problem

00:03 A 'regular' root is of the order 2

00:08 When we have a number (A) in a root of the order (B) in a root of the order (C)

00:16 The result equals the number (A) in a root of the order of their product (B times C)

00:21 We will apply this formula to our exercise

00:26 Let's calculate the of order multiplication

00:37 When we have a number (A) to the power of (B) in a root of the order (C)

00:41 The result equals number (A) to the power of their quotient (B divided by C)

00:44 We will apply this formula to our exercise

00:53 This is the 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 $2$ .