To simplify the expression \((10\sqrt{256})^5 / 512^{1/3}\), we will first simplify each part step by step.
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Simplify \(\sqrt{256}\): \[ \sqrt{256} = 16 \]
So, we can rewrite the expression: \[ (10\sqrt{256})^5 = (10 \cdot 16)^5 = (160)^5 \]
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Calculate \((160)^5\): At this stage, we can just keep it as \((160)^5\) for now.
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Simplify \(512^{1/3}\): We can rewrite \(512\) as a power of \(2\): \[ 512 = 2^9 \] Thus, \[ 512^{1/3} = (2^9)^{1/3} = 2^{9/3} = 2^3 = 8. \]
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Put it all together: Now we substitute everything back into the original expression: \[ \frac{(160)^5}{8} \]
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Simplify the expression: We want to simplify \(\frac{(160)^5}{8}\): \[ 160 = 8 \cdot 20 \] Thus, \[ (160)^5 = (8 \cdot 20)^5 = 8^5 \cdot 20^5 \]
Now substituting this in gives us: \[ \frac{8^5 \cdot 20^5}{8} = \frac{8^5}{8} \cdot 20^5 = 8^{5-1} \cdot 20^5 = 8^4 \cdot 20^5 \]
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Final Answer: We can write this final expression as: \[ 8^4 \cdot 20^5 \]
Thus, an equivalent expression for \((10\sqrt{256})^5/512^{1/3}\) is:
\[ 8^4 \cdot 20^5 \]
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