To simplify the expression \(\sqrt{20p^5q^{10}}\), we can break it down into its components.
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Factor the expression inside the square root:
\[ \sqrt{20p^5q^{10}} = \sqrt{20} \cdot \sqrt{p^5} \cdot \sqrt{q^{10}} \]
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Simplify each component:
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For \(\sqrt{20}\): \[ 20 = 4 \cdot 5 \quad \text{thus} \quad \sqrt{20} = \sqrt{4 \cdot 5} = \sqrt{4} \cdot \sqrt{5} = 2\sqrt{5} \]
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For \(\sqrt{p^5}\): \[ p^5 = p^4 \cdot p = (p^2)^2 \cdot p \quad \text{thus} \quad \sqrt{p^5} = \sqrt{(p^2)^2 \cdot p} = p^2\sqrt{p} \]
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For \(\sqrt{q^{10}}\): \[ q^{10} = (q^5)^2 \quad \text{thus} \quad \sqrt{q^{10}} = q^5 \]
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Combine the results:
Now putting it all together: \[ \sqrt{20p^5q^{10}} = 2\sqrt{5} \cdot p^2\sqrt{p} \cdot q^5 \]
Rearranging gives us: \[ = 2p^2q^5\sqrt{5p} \]
Thus, the simplified form of \(\sqrt{20p^5q^{10}}\) is: \[ \boxed{2p^2q^5\sqrt{5p}} \]