Simplify the Nested Square Root: √(√(16x²))

Video Solution

Solution Steps

00:00 Solve the following problem

00:05 When we have a root of the multiplication (A times B)

00:09 We can write it as a multiplication of the root of each term

00:12 We'll apply this formula to our exercise and proceed to break down the root

00:18 Break down 16 to 4 squared

00:22 The root of any number (A) squared cancels the square

00:30 Apply this formula to our exercise and cancel the squares

00:49 Break down 4 to 2 squared

00:53 The root cancels the square

00:56 This is the solution

Step-by-Step Solution

To solve the expression $\sqrt{\sqrt{16 \cdot x^2}}$ , follow these steps:

Step 1: Simplify the innermost root $\sqrt{16 \cdot x^2}$ .
Here, apply $\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$ . Thus, $\sqrt{16 \cdot x^2} = \sqrt{16} \cdot \sqrt{x^2}$ .
Step 2: Calculate each component:
- $\sqrt{16} = 4$ because $16^{1/2} = 4$ .
- $\sqrt{x^2} = x$ assuming $x$ is non-negative.
Step 3: Combine results from Step 2: $\sqrt{16} \cdot \sqrt{x^2} = 4 \cdot x = 4x$ .
Step 4: Simplify the outer square root: $\sqrt{4x}$ .
Applying $\sqrt{4x} = \sqrt{4} \cdot \sqrt{x}$ , we have $\sqrt{4} = 2$ .
Thus, $\sqrt{4x} = 2 \cdot \sqrt{x} = 2\sqrt{x}$ .

Therefore, the simplified form of $\sqrt{\sqrt{16 \cdot x^2}}$ is $2\sqrt{x}$ . This corresponds to choice 1.