Solve the Nested Square Root: Simplifying √(√8)

Video Solution

Solution Steps

00:05 Let's solve this problem together.

00:08 Remember, a regular root is usually of order two.

00:13 Imagine we have a number, A, under a root of order B, then again under a root of order C.

00:20 The result is the number, A, under a root. This new root's order is the product of B and C.

00:28 Let's apply this formula to our problem now.

00:31 First, we calculate the multiplication of the orders.

00:37 If we have a number, A, raised to the power of B, under a root of order C.

00:43 Then, the result is A raised to the power of B divided by C.

00:49 Let's apply this approach to our problem again.

00:53 And there you have the solution! Great job tackling this problem!

Step-by-Step Solution

In order to solve the given problem, we'll follow these steps:

Step 1: Convert the inner square root to an exponent: $\sqrt{8} = 8^{1/2}$ .
Step 2: Apply the root of a root property: $\sqrt{\sqrt{8}} = (\sqrt{8})^{1/2} = (8^{1/2})^{1/2}$ .
Step 3: Simplify the expression using exponent rules: $(8^{1/2})^{1/2} = 8^{(1/2) \cdot (1/2)} = 8^{1/4}$ .

The nested root expression simplifies to $8^{1/4}$ .

Therefore, the simplified expression of $\sqrt{\sqrt{8}}$ is $8^{\frac{1}{4}}$ .

After comparing this result with the multiple choice answers, choice 2 is correct.