To simplify the expression \((256 \sqrt{10})^5 / 512^{1/3}\), we can start by breaking down the components step by step.
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Simplify \( (256 \sqrt{10})^5 \):
- \(256\) can be expressed as \(2^8\) since \(256 = 2^8\).
- The square root of \(10\) can be expressed as \(10^{1/2}\).
- Therefore, we rewrite: \[ 256 \sqrt{10} = 2^8 \cdot 10^{1/2} \]
- Now we apply the exponent: \[ (256 \sqrt{10})^5 = (2^8 \cdot 10^{1/2})^5 = 2^{8 \cdot 5} \cdot 10^{(1/2) \cdot 5} = 2^{40} \cdot 10^{5/2} \]
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Simplify \(512^{1/3}\):
- \(512\) can be expressed as \(2^9\) since \(512 = 2^9\).
- Now we can simplify: \[ 512^{1/3} = (2^9)^{1/3} = 2^{9/3} = 2^{3} = 8 \]
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Rewrite the original expression using the simplified forms: \[ \frac{(256 \sqrt{10})^5}{512^{1/3}} = \frac{2^{40} \cdot 10^{5/2}}{2^{3}} \]
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Combine the exponents of \(2\):
- We subtract the exponents in the numerator and the denominator: \[ 2^{40} / 2^{3} = 2^{40 - 3} = 2^{37} \]
- Thus, we have: \[ \frac{(256 \sqrt{10})^5}{512^{1/3}} = 2^{37} \cdot 10^{5/2} \]
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Express \(10^{5/2}\) in terms of radicals:
- We can rewrite \(10^{5/2}\) as: \[ 10^{5/2} = (10^5)^{1/2} = \sqrt{10^5} = \sqrt{10^4 \cdot 10} = 10^2 \cdot \sqrt{10} = 100 \sqrt{10} \]
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Final expression:
- Therefore, the entire expression can be combined and simplified to: \[ 2^{37} \cdot 10^{5/2} = 2^{37} \cdot 100 \sqrt{10} \]
So the equivalent expression is:
\[ \boxed{2^{37} \cdot 100 \sqrt{10}} \]
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