using log table evaluate
`root(5)(8.0125)`

Answer

To evaluate \( \sqrt[5]{8.0125} \) using logarithms, we can follow these steps:

### Step 1: Rewrite the expression using logarithms
We start with the expression:
\[ x = \sqrt[5]{8.0125} \]
This can be rewritten using exponents:
\[ x = 8.0125^{\frac{1}{5}} \]

### Step 2: Take the logarithm of both sides
Now, we take the logarithm of both sides:
\[ \log x = \log(8.0125^{\frac{1}{5}}) \]

### Step 3: Use the power rule of logarithms
Using the power rule of logarithms, we can bring down the exponent:
\[ \log x = \frac{1}{5} \log(8.0125) \]

### Step 4: Look up the logarithm value in the log table
Next, we need to find \( \log(8.0125) \) using a logarithm table. From the log table, we find:
\[ \log(8.0125) \approx 0.90377 \]

### Step 5: Substitute the logarithm value back into the equation
Now, we substitute this value back into our equation:
\[ \log x = \frac{1}{5} \times 0.90377 \]

### Step 6: Calculate the value
Calculating this gives:
\[ \log x = 0.180754 \]

### Step 7: Find the antilogarithm
To find \( x \), we take the antilogarithm:
\[ x = 10^{0.180754} \]

### Step 8: Calculate the antilogarithm
Using a calculator or log table, we find:
\[ x \approx 1.517 \]

Thus, the value of \( \sqrt[5]{8.0125} \) is approximately \( 1.517 \).

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