Solve the Nested Radical: Cube Root of Square Root of 64

Video Solution

Solution Steps

00:00 Solve the following problem

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

00:09 When we have a number (A) in a root of the order (B) in a root of the order (C)

00:14 The result equals the number (A) in a root of the order of their product (B times C)

00:18 We will apply this formula to our exercise

00:23 Let's calculate the order of multiplication

00:30 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

00:46 This is the solution

Step-by-Step Solution

To solve this problem, we'll express the nested roots using exponents and then simplify:

We start with the inner expression:

$\sqrt{64}$ is equivalent to $64^{1/2}$ .

Next, apply the cube root:

$\sqrt[3]{64^{1/2}}$ is equivalent to $(64^{1/2})^{1/3}$ .

Using properties of exponents, we simplify the expression:

$(64^{1/2})^{1/3} = 64^{(1/2) \cdot (1/3)} = 64^{1/6}$ .

Now, evaluate $64^{1/6}$ :

Since $64 = 2^6$ , we have:

$64^{1/6} = (2^6)^{1/6} = 2^{6 \cdot (1/6)} = 2^{1} = 2$ .

Therefore, the solution to the exercise $\sqrt[3]{\sqrt{64}}$ is $\mathbf{2}$ .