To simplify the expression \(\sqrt{80x^3y^2}\), we can break it down as follows:
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Factor out the constant and the variables:
- Start by breaking down \(80\): \[ 80 = 16 \times 5 = 4^2 \times 5 \]
- The variables are already in exponent form.
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Rewrite the expression: \[ \sqrt{80x^3y^2} = \sqrt{16 \cdot 5 \cdot x^3 \cdot y^2} = \sqrt{16} \cdot \sqrt{5} \cdot \sqrt{x^3} \cdot \sqrt{y^2} \]
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Simplify each square root:
- \(\sqrt{16} = 4\)
- \(\sqrt{5}\) remains as \(\sqrt{5}\)
- For \(\sqrt{x^3}\): \[ \sqrt{x^3} = \sqrt{x^2 \cdot x} = \sqrt{x^2} \cdot \sqrt{x} = x \sqrt{x} \]
- For \(\sqrt{y^2}\): \[ \sqrt{y^2} = y \]
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Combine everything: \[ \sqrt{80x^3y^2} = 4 \cdot \sqrt{5} \cdot x \sqrt{x} \cdot y = 4xy \sqrt{5x} \]
Thus, the simplified form of \(\sqrt{80x^3y^2}\) is \(4xy\sqrt{5x}\).
Final answer: \(4xy\sqrt{5x}\)
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