To simplify \(\sqrt{20x^8}\), we first break it down into its components:
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Factor 20: \[ 20 = 4 \times 5 \] Here, \(4\) is a perfect square.
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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:
\[ \boxed{2x^4 \sqrt{5}} \]
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