To solve the expression \( \sqrt[3]{\frac{0.0072 \times 81^2}{\sqrt{23140}}} \), we will simplify step by step.
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Calculate \( 81^2 \): \[ 81^2 = 6561 \]
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Calculate \( 0.0072 \times 6561 \): \[ 0.0072 \times 6561 = 47.3152 \]
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Calculate \( \sqrt{23140} \): Approximating \( \sqrt{23140} \): Since \( 23140 \approx 23136 \) which is \( 152^2 \), we can calculate that \( \sqrt{23140} \approx 152.0 \).
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Divide \( 47.3152 \) by \( \sqrt{23140} \): \[ \frac{47.3152}{\sqrt{23140}} \approx \frac{47.3152}{152.0} \approx 0.311 \]
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Calculate the cube root: \[ \sqrt[3]{0.311} \approx 0.68 \]
So, the final answer is approximately:
\[ \sqrt[3]{\frac{0.0072 \times 81^2}{\sqrt{23140}}} \approx 0.68 \]
If you need a more exact value, we can use a calculator for the cube root of \( 0.311 \). The nearest simpler form gives you an estimate that should suffice for practical purposes.