To evaluate the square root of \( 0.000562 \) using logarithmic tables, we can follow these steps:
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Convert to Scientific Notation: \[ 0.000562 = 5.62 \times 10^{-4} \]
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Logarithm of the Number: Next, we find the logarithm of \( 5.62 \). If we can access a logarithm table, we look up \( \log_{10}(5.62) \).
From standard logarithm tables or calculators: \[ \log_{10}(5.62) \approx 0.75 \]
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Adding Logarithm of the Power of Ten: Since we have \( 5.62 \times 10^{-4} \): \[ \log_{10}(0.000562) = \log_{10}(5.62) + \log_{10}(10^{-4}) \ = \log_{10}(5.62) - 4 \] Substituting the value we found: \[ \log_{10}(0.000562) \approx 0.75 - 4 = -3.25 \]
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Finding the Square Root: The square root can be found by halving the logarithm: \[ \log_{10}(\sqrt{0.000562}) = \frac{1}{2} \log_{10}(0.000562) \ = \frac{1}{2} \cdot (-3.25) = -1.625 \]
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Finding the antilogarithm: Now we need to find \( 10^{-1.625} \). This may also require a logarithm table, or we can calculate it as: \[ 10^{-1.625} \approx 0.0234 \]
Thus, the square root of \( 0.000562 \) is approximately: \[ \sqrt{0.000562} \approx 0.0234 \]