To evaluate [tex]\(\sqrt[4]{\frac{0.0763}{309 \times 0.008465}}\)[/tex] correct to 3 significant figures using logarithm tables, follow these steps:
1. Calculate the product in the denominator:
First, we need to multiply the two numbers in the denominator:
[tex]\[ 309 \times 0.008465 \][/tex]
2. Find the fraction:
Next, we'll divide the numerator by the result of the multiplication:
[tex]\[ \frac{0.0763}{309 \times 0.008465} \][/tex]
3. Use logarithm tables:
- Look up the logarithm of the numerator (0.0763).
- Look up the logarithm of the result from the denominator.
- Subtract the logarithm of the denominator from the logarithm of the numerator.
Let's denote:
[tex]\[ \log_{10}(0.0763) = a \][/tex]
[tex]\[ \log_{10}(309 \times 0.008465) = b \][/tex]
So, the logarithm of the fraction is:
[tex]\[ \log_{10}\left(\frac{0.0763}{309 \times 0.008465}\right) = a - b \][/tex]
4. Find the fourth root:
To find the fourth root, divide the resulting logarithm by 4:
[tex]\[ \log_{10}\left(\sqrt[4]{\frac{0.0763}{309 \times 0.008465}}\right) = \frac{a - b}{4} \][/tex]
5. Convert back from logarithms:
Use the antilogarithm table or a calculator to find the actual value from its logarithm. Let [tex]\( c \)[/tex] be:
[tex]\[ c = \frac{a - b}{4} \][/tex]
So, the final result is:
[tex]\[ \sqrt[4]{\frac{0.0763}{309 \times 0.008465}} = 10^c \][/tex]
6. Round to 3 significant figures:
The final result of the fourth root should be rounded to three significant figures.
From previous steps and evaluations, we get:
- The value of the fraction is approximately [tex]\(0.02917017913089688\)[/tex].
- The fourth root of this fraction is approximately [tex]\(0.413\)[/tex].
Hence, the value of [tex]\(\sqrt[4]{\frac{0.0763}{309 \times 0.008465}} \approx 0.413\)[/tex] to 3 significant figures.
