Certainly! Let's evaluate the expression using logarithms:
[tex]\[
\frac{(0.08362)^2}{\sqrt[3]{0.07185}}
\][/tex]
### Step 1: Express the given values in terms of logarithms
Given:
[tex]\[ a = 0.08362 \][/tex]
[tex]\[ b = 0.07185 \][/tex]
We need to evaluate:
[tex]\[ \frac{a^2}{b^{1/3}} \][/tex]
### Step 2: Apply logarithms to the numerator
The numerator is:
[tex]\[ a^2 \][/tex]
Using logarithm properties:
[tex]\[ \log(a^2) = 2 \log(a) \][/tex]
### Step 3: Apply logarithms to the denominator
The denominator is:
[tex]\[ b^{1/3} \][/tex]
Using logarithm properties:
[tex]\[ \log(b^{1/3}) = \frac{1}{3} \log(b) \][/tex]
### Step 4: Combine the parts using logarithm properties
We have the expression:
[tex]\[ \log\left(\frac{a^2}{b^{1/3}}\right) = \log(a^2) - \log(b^{1/3}) \][/tex]
Substitute the logarithm expressions from Steps 2 and 3:
[tex]\[ \log\left(\frac{a^2}{b^{1/3}}\right) = 2 \log(a) - \frac{1}{3} \log(b) \][/tex]
### Step 5: Substitute the values of [tex]\(\log(a)\)[/tex] and [tex]\(\log(b)\)[/tex]
From the numerical results, we know:
[tex]\[ \log(0.08362) = -2.481472553053718 \][/tex]
[tex]\[ \log(0.07185) = -2.633174666457103 \][/tex]
Now, calculate each term:
[tex]\[ 2 \log(0.08362) = 2 \times -2.481472553053718 = -4.962945106107436 \][/tex]
[tex]\[ \frac{1}{3} \log(0.07185) = \frac{1}{3} \times -2.633174666457103 = -0.8777248888190343 \][/tex]
Combine these results:
[tex]\[ -4.962945106107436 - (-0.8777248888190343) = -4.962945106107436 + 0.8777248888190343 = -4.085220217288402 \][/tex]
### Step 6: Convert back from the logarithm
To find the original value, we take the exponential:
[tex]\[ \exp(-4.085220217288402) \approx 0.01681943497773546 \][/tex]
Thus, the evaluated result of the expression [tex]\(\frac{(0.08362)^2}{\sqrt[3]{0.07185}}\)[/tex] is:
[tex]\[
\boxed{0.01681943497773546}
\][/tex]
