Solve Nested Cube Roots: Simplifying ∛(∛512)

Q: How do I recognize that 512 is a perfect cube?

Look for patterns in powers of small numbers. Since $ 2^9 = 512 $ and 9 is divisible by 3, we know $ \sqrt[3]{512} = \sqrt[3]{2^9} = 2^3 = 8 $.

Q: Is there a pattern for nested cube roots?

Yes! For $ \sqrt[3]{\sqrt[3]{a}} $, if $ a = b^9 $, then the answer is simply $ b $. This is because $ (b^3)^{1/3} = b $.

Q: How can I check my answer is correct?

Work backwards! If your answer is 2, check: $ 2^3 = 8 $, then $ 8^3 = 512 $. Since we get back to 512, our answer is correct!

Nested Radical Expressions with Cube Roots

Solve the following exercise:

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

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

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!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following exercise:

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

Step-by-step 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:

$\sqrt[3]{512} = \sqrt[3]{2^9} = 2^{9/3} = 2^3 = 8$ .

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

$\sqrt[3]{8} = \sqrt[3]{2^3} = 2^{3/3} = 2$ .

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

Final Answer

Key Points to Remember

Essential concepts to master this topic

Rule: Simplify innermost radical first, then work outward systematically
Technique: Use exponent rules: $\sqrt[3]{2^9} = 2^{9/3} = 2^3 = 8$
Check: Verify $2^3 = 8$ and $8^3 = 512$ ✓

Common Mistakes

Avoid these frequent errors

Trying to combine the radicals without simplifying first
Don't write $\sqrt[3]{\sqrt[3]{512}} = \sqrt[9]{512}$ = wrong approach! This creates confusion and doesn't use the step-by-step nature of nested operations. Always simplify the innermost radical completely before moving to the outer one.

Practice Quiz

Test your knowledge with interactive questions

Solve the following exercise:

$\sqrt[10]{\sqrt[10]{1}}=$

FAQ

Everything you need to know about this question

Why can't I just find the 9th root of 512 directly?

While mathematically equivalent, this approach is much harder! Nested radicals are designed to be solved step-by-step from inside out, making the computation easier.

How do I recognize that 512 is a perfect cube?

Look for patterns in powers of small numbers. Since $2^9 = 512$ and 9 is divisible by 3, we know $\sqrt[3]{512} = \sqrt[3]{2^9} = 2^3 = 8$ .

What if the inner number wasn't a perfect cube?

You'd still simplify as much as possible first! For example, $\sqrt[3]{16} = \sqrt[3]{2^4} = 2^{4/3} = 2 \cdot 2^{1/3}$ , then work with that result in the outer radical.

Is there a pattern for nested cube roots?

Yes! For $\sqrt[3]{\sqrt[3]{a}}$ , if $a = b^9$ , then the answer is simply $b$ . This is because $(b^3)^{1/3} = b$ .

How can I check my answer is correct?

Work backwards! If your answer is 2, check: $2^3 = 8$ , then $8^3 = 512$ . Since we get back to 512, our answer is correct!

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