To simplify the expression \((10\sqrt{256})^5 / 512^{1/3}\), we will start with simplifying each part of the expression.
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Calculate \( \sqrt{256} \): \[ \sqrt{256} = 16 \] Therefore, we have: \[ 10\sqrt{256} = 10 \times 16 = 160 \]
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Now, raise \(160\) to the power of \(5\): \[ (10\sqrt{256})^5 = 160^5 \]
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Now, simplify \(512^{1/3}\): First, we note that \(512\) can be expressed as \(2^9\) (because \(2^9 = 512\)). Therefore: \[ 512^{1/3} = (2^9)^{1/3} = 2^{9/3} = 2^3 = 8 \]
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Put it all together: Now we can rewrite the original expression: \[ \frac{160^5}{512^{1/3}} = \frac{160^5}{8} \]
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Evaluate \(160^5/8\): We can express \(160\) in terms of its prime factors: \[ 160 = 2^5 \times 5^1 \] Therefore: \[ 160^5 = (2^5 \times 5^1)^5 = 2^{25} \times 5^5 \]
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Now simplify \(\frac{160^5}{8}\): Since \(8 = 2^3\): \[ \frac{160^5}{8} = \frac{2^{25} \times 5^5}{2^3} = 2^{25 - 3} \times 5^5 = 2^{22} \times 5^5 \]
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Final expression: Thus, the equivalent expression for \((10\sqrt{256})^5 / 512^{1/3}\) is: \[ 2^{22} \times 5^5 \]
This can also be written in exponential form or simplified further, but this is a valid equivalent expression: \[ \boxed{2^{22} \times 5^5} \]
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