Simplify the Nested Radical: ∛(√(64x¹²))

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) to the power of (B) in a root of order (C)

00:13 The result equals the number (A) to the power of their product (B times C)

00:17 We'll apply this formula to our exercise, and proceed to calculate the product of the orders

00:31 When we have a root of a product (A times B)

00:36 We can write it as a product of the root of each term

00:39 We'll apply this formula to our exercise, and break down the root

00:44 Break down 64 to 2 to the power of 6

00:50 When we have a number (A) to the power of (B) in a root of order (C)

00:55 The result equals number (A) to the power of their quotient (B divided by C)

01:01 We'll apply this formula to our exercise, and proceed to calculate the quotient of powers

01:11 This is the solution

Step-by-Step Solution

To solve the problem $\sqrt[3]{\sqrt{64 \cdot x^{12}}}$ , follow these detailed steps:

Step 1: Simplify the inner expression.
The expression inside the radical is $64 \cdot x^{12}$ .
Step 2: Simplify the inner square root.

First, we need to find $\sqrt{64 \cdot x^{12}}$ .

The square root of a product can be expressed as the product of the square roots: $\sqrt{64} \cdot \sqrt{x^{12}}$ .

Simplifying further, we find:
- $\sqrt{64} = 8$ , since $8^2 = 64$ .
- $\sqrt{x^{12}} = x^{6}$ , because $(x^{6})^2 = x^{12}$ .
Thus, the inner square root becomes $8x^6$ .
Step 3: Simplify using the cube root.

Next, apply the cube root to the result of the inner square root: $\sqrt[3]{8x^6}$ .

The cube root of a product can also be expressed as the product of the cube roots:
- $\sqrt[3]{8} = 2$ , since $2^3 = 8$ .
- $\sqrt[3]{x^6} = x^{6/3} = x^{2}$ , because $(x^2)^3 = x^6$ .
Thus, the expression simplifies to $2x^2$ .

Therefore, the solution to this problem is $2x^2$ , which corresponds to choice 2 in the provided options.