Solve Nested Cube Roots: Simplifying ∛(∛512)

Question

Solve the following exercise:

$\sqrt[3]{\sqrt[3]{512}}=$

00:00 Solve the following problem

00:03 When we have a number (A) with the root order (B) with the root order (C)

00:08 The result equals the number (A) with the root order of their product (B×C)

00:16 Let's apply this formula to our exercise

00:22 Calculate the order multiplication

00:29 Break down 512 to 2 to the power of 9

00:33 When we have a number (A) to the power of (B) with the root order (B²)

00:37 The root and the power cancel each other out and the result is A

00:40 Let's apply this formula to our exercise

00:44 This is the solution

To solve this problem, we'll proceed with the following steps:

Step 1: Compute $\sqrt[3]{512}$ .
Given $512$ , recognize that $512 = 2^9$ since $2^9 = 512$ . Thus, we have:

Step 2: Compute $\sqrt[3]{8}$ .
From Step 1, we found $\sqrt[3]{512} = 8$ . Now find $\sqrt[3]{8}$ :

Therefore, the solution to the expression $\sqrt[3]{\sqrt[3]{512}}$ is $2$ .