Question

Simplify. Remove all the perfect squares from the root (sqrt) 20x^8

Answer 1

To simplify \(\sqrt{20x^8}\), we first break it down into its components:

1. **Factor 20**:
\[
20 = 4 \times 5
\]
Here, \(4\) is a perfect square.

2. **Analyze \(x^8\)**:
\[
x^8 = (x^4)^2
\]
This means \(x^8\) is also a perfect square.

Now, we can rewrite \(\sqrt{20x^8}\) using these factors:

\[
\sqrt{20x^8} = \sqrt{4 \times 5 \times (x^4)^2}
\]

Next, we can separate the square root of the perfect squares from the square root of the non-perfect squares:

\[
\sqrt{20x^8} = \sqrt{4} \times \sqrt{5} \times \sqrt{(x^4)^2}
\]

Calculating the square roots of the perfect squares:

\[
\sqrt{4} = 2 \quad \text{and} \quad \sqrt{(x^4)^2} = x^4
\]

Putting it all together, we have:

\[
\sqrt{20x^8} = 2 \times x^4 \times \sqrt{5}
\]

Thus, the simplified expression is:

\[
2x^4 \sqrt{5}
\]

Therefore, the final answer is:

\[
\boxed{2x^4 \sqrt{5}}
\]