To evaluate \( \frac{\sqrt[3]{1.6540}}{45.73 \times 0.56} \) using logarithm tables, follow these steps:
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Calculate the logarithm of each component.
a. Cube root of 1.6540: - The logarithm of \( 1.6540 \) can be found from log tables. - \(\log_{10}(1.6540) \approx 0.219\) (the exact value may vary a bit based on the log table used). - To find the logarithm of the cube root, use the property: \[ \log_{10}(\sqrt[3]{x}) = \frac{1}{3} \log_{10}(x) \] - Therefore, \[ \log_{10}(\sqrt[3]{1.6540}) \approx \frac{1}{3} \times 0.219 \approx 0.073. \]
b. Calculate logarithms of 45.73 and 0.56: - \(\log_{10}(45.73) \approx 1.660\). - \(\log_{10}(0.56) \approx -0.253\).
c. Combine logarithms for the denominator: - Since we need to calculate \(45.73 \times 0.56\), \[ \log_{10}(45.73 \times 0.56) = \log_{10}(45.73) + \log_{10}(0.56) \approx 1.660 - 0.253 = 1.407. \]
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Find the logarithm of the fraction: \[ \log_{10}\left(\frac{\sqrt[3]{1.6540}}{45.73 \times 0.56}\right) = \log_{10}(\sqrt[3]{1.6540}) - \log_{10}(45.73 \times 0.56) \] \[ \approx 0.073 - 1.407 = -1.334. \]
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Calculate the antilogarithm:
- The antilogarithm of \(-1.334\) can be computed as: \[ 10^{-1.334} \approx 0.0464. \]
Thus, the value of \( \frac{\sqrt[3]{1.6540}}{45.73 \times 0.56} \) is approximately \( 0.0464 \).