Solve Nested Cube Roots: Simplifying ∛(∛512)

Question

Solve the following exercise:

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

00:06 Let's solve this problem together.

00:09 Imagine we have a number A under a root with order B, then under another root with order C.

00:15 This equals number A, under one root with order B times C. Pretty neat, right?

00:22 Alright, let's use this formula in our exercise.

00:28 First, we need to multiply the orders.

00:35 Now, let's break down 512. It's two to the power of nine.

00:40 Remember, if we have a number A to the power of B with root order B squared…

00:46 The root and power cancel out, leaving us with just A.

00:50 Let's apply this in our exercise and see it work.

00:53 And there you have it! That's the solution. Great job!

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 $2$ .