To simplify the expression \((10\sqrt{256})^5 / 512^{1/3}\), we will work with both the numerator and the denominator separately.
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Simplify the numerator \((10\sqrt{256})^5\):
- First, find \(\sqrt{256}\): \[ \sqrt{256} = 16. \]
- Now, substitute back into the expression: \[ 10\sqrt{256} = 10 \times 16 = 160. \]
- Now raise \(160\) to the power of \(5\): \[ (160)^5. \]
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Simplify the denominator \(512^{1/3}\):
- First, express \(512\) as a power of \(2\): \[ 512 = 2^9. \]
- Now calculate \(2^9\) raised to the power of \(\frac{1}{3}\): \[ 512^{1/3} = (2^9)^{1/3} = 2^{9/3} = 2^3 = 8. \]
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Combine the simplified forms: Now we have: \[ \frac{(160)^5}{8}. \]
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Further simplification can be performed:
- Write \(160\) as \(2^5 \times 5\): \[ 160 = 2^5 \times 5. \]
- Now raise this to the \(5\): \[ (160)^5 = (2^5 \times 5)^5 = 2^{25} \times 5^5. \]
- Therefore, the expression now looks like: \[ \frac{2^{25} \times 5^5}{8}. \]
- Replace \(8\) with \(2^3\): \[ \frac{2^{25} \times 5^5}{2^3} = 2^{25-3} \times 5^5 = 2^{22} \times 5^5. \]
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Final Equivalent Expression: The equivalent expression for \((10\sqrt{256})^5 / 512^{1/3}\) is: \[ 2^{22} \times 5^5. \]