using log table evaluate`root(7)(142.71)`

Answer

To evaluate \( \sqrt[7]{142.71} \) using logarithms, we can follow these steps:

### Step 1: Rewrite the Expression
We start by rewriting the expression \( \sqrt[7]{142.71} \) in terms of logarithms:
\[ x = 142.71^{\frac{1}{7}} \]

### Step 2: Take the Logarithm
Now, we take the logarithm of both sides:
\[ \log x = \log(142.71^{\frac{1}{7}}) \]

### Step 3: Apply Logarithmic Identity
Using the logarithmic identity \( \log(m^n) = n \cdot \log m \), we can simplify:
\[ \log x = \frac{1}{7} \log(142.71) \]

### Step 4: Find the Logarithm Value
Next, we need to find \( \log(142.71) \) using a logarithm table. From the table, we find:
\[ \log(142.71) \approx 2.1549 \]

### Step 5: Substitute and Calculate
Now we substitute this value back into our equation:
\[ \log x = \frac{1}{7} \cdot 2.1549 \]
Calculating this gives:
\[ \log x \approx 0.3077 \]

### Step 6: Find the Antilogarithm
To find \( x \), we take the antilogarithm:
\[ x = 10^{0.3077} \]

### Step 7: Calculate the Antilogarithm
Using a calculator or logarithm table, we find:
\[ x \approx 2.03 \]

### Final Answer
Thus, the value of \( \sqrt[7]{142.71} \) is approximately:
\[ \boxed{2.03} \]

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