To simplify \(\sqrt{54x^7}\), we start by breaking down the expression under the square root into its prime factors and separating the perfect squares from the non-perfect squares.
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Factor 54: \[ 54 = 2 \times 27 = 2 \times 3^3 \]
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Rewrite the expression: \[ 54x^7 = 2 \times 3^3 \times x^7 \]
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Identify perfect squares:
- From \(3^3\), we can extract \(3^2\) as a perfect square: \[ 3^3 = 3^2 \times 3^1 \]
- For \(x^7\), we can extract \(x^6\) as a perfect square: \[ x^7 = x^6 \times x^1 \]
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Put it all together inside the square root: \[ \sqrt{54x^7} = \sqrt{(2 \times 3^2 \times x^6) \times (3^1 \times x^1)} = \sqrt{(2 \times 9 \times x^6) \times (3 \times x)} \]
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Use the property of square roots to separate perfect squares from non-perfect squares: \[ \sqrt{54x^7} = \sqrt{2} \cdot \sqrt{9} \cdot \sqrt{x^6} \cdot \sqrt{3} \cdot \sqrt{x} \]
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Calculate the square roots of the perfect squares:
- \(\sqrt{9} = 3\)
- \(\sqrt{x^6} = x^3\)
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Combine the simplified parts: \[ \sqrt{54x^7} = 3x^3\sqrt{2 \times 3 \times x} = 3x^3\sqrt{6x} \]
Thus, the simplified form is: \[ \sqrt{54x^7} = 3x^3\sqrt{6x} \]