To simplify the expression \((256\sqrt{10})^{5}/(512^{1/3})\), we can break it down step-by-step.
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Rewrite \(256\) and \(512\) as powers of \(2\):
- \(256 = 2^8\)
- \(512 = 2^9\)
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Rewrite \(\sqrt{10}\):
- \(\sqrt{10} = 10^{1/2}\)
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Rewriting the original expression: \[ (256 \sqrt{10})^5 = (2^8 \cdot 10^{1/2})^5 = (2^8)^5 \cdot (10^{1/2})^5 = 2^{40} \cdot 10^{5/2} \]
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Now rewrite \(512^{1/3}\): \[ 512^{1/3} = (2^9)^{1/3} = 2^{9/3} = 2^3 = 8 \]
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Now substitute into our expression: \[ \frac{(256 \sqrt{10})^5}{512^{1/3}} = \frac{2^{40} \cdot 10^{5/2}}{8} = \frac{2^{40} \cdot 10^{5/2}}{2^3} = 2^{40-3} \cdot 10^{5/2} = 2^{37} \cdot 10^{5/2} \]
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Final expression: \[ 2^{37} \cdot 10^{5/2} \]
Thus, the equivalent expression for \((256\sqrt{10})^5/512^{1/3}\) is:
\[ 2^{37} \cdot 10^{5/2} \]
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