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

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 order (B) in a root of order (C)

00:12 The result equals the number (A) in a root of order of their product (B times C)

00:17 We will apply this formula to our exercise

00:23 Let's calculate the order multiplication

00:32 When we have a number (A) to the power of (B) in a root of the order (C)

00:36 The result equals the number (A) to the power of their quotient (B divided by C)

00:39 We will apply this formula to our exercise

00:42 This is the solution

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.