To simplify the expression \( x^3 \sqrt{147} - 2x \sqrt{27} x^4 \), let's work through it step by step.
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Simplify \(\sqrt{147}\): \[ \sqrt{147} = \sqrt{49 \times 3} = \sqrt{49} \cdot \sqrt{3} = 7\sqrt{3} \]
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Simplify \(\sqrt{27}\): \[ \sqrt{27} = \sqrt{9 \times 3} = \sqrt{9} \cdot \sqrt{3} = 3\sqrt{3} \]
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Substituting back into the expression: Now substitute these simplifications back into the original expression: \[ x^3 \sqrt{147} - 2x \sqrt{27} x^4 = x^3 (7\sqrt{3}) - 2x(3\sqrt{3}) x^4 \]
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Distributing and simplifying: For the first term, \[ x^3 (7\sqrt{3}) = 7x^3\sqrt{3} \] For the second term, \[ 2x(3\sqrt{3}) x^4 = 6x^5\sqrt{3} \]
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Combining the terms: Now, rewrite the entire expression: \[ 7x^3\sqrt{3} - 6x^5\sqrt{3} \]
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Factor out the common term: Factor out \(\sqrt{3}\): \[ \sqrt{3}(7x^3 - 6x^5) \]
The final simplified expression is: \[ \sqrt{3}(7x^3 - 6x^5) \]
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