Find the value of the `root(5)(242)` correct to 4 decimal places

Answer

To find the value of \( \sqrt[5]{242} \) correct to four decimal places, we can follow these steps:

### Step 1: Rewrite the expression
We can express \( \sqrt[5]{242} \) as \( 242^{1/5} \).

### Step 2: Factor 242
Next, we can factor 242 to make calculations easier. We know that:
\[ 242 = 2 \times 121 = 2 \times 11^2 \]
Thus, we can rewrite \( 242^{1/5} \) as:
\[ 242^{1/5} = (2 \times 11^2)^{1/5} = 2^{1/5} \times 11^{2/5} \]

### Step 3: Calculate \( 2^{1/5} \) and \( 11^{2/5} \)
We can find \( 2^{1/5} \) and \( 11^{2/5} \) using a calculator or by approximation.

1. **Calculating \( 2^{1/5} \)**:
- Using a calculator, we find:
\[ 2^{1/5} \approx 1.1487 \]

2. **Calculating \( 11^{2/5} \)**:
- First, calculate \( 11^{1/5} \):
\[ 11^{1/5} \approx 1.6197 \]
- Then square this value:
\[ 11^{2/5} \approx (1.6197)^2 \approx 2.6210 \]

### Step 4: Multiply the results
Now we multiply \( 2^{1/5} \) and \( 11^{2/5} \):
\[ 242^{1/5} \approx 1.1487 \times 2.6210 \]

Calculating this gives:
\[ 242^{1/5} \approx 3.0158 \]

### Step 5: Round to four decimal places
Finally, we round the result to four decimal places:
\[ \sqrt[5]{242} \approx 3.0158 \]

### Final Answer
Thus, the value of \( \sqrt[5]{242} \) correct to four decimal places is:
\[ \boxed{3.0158} \]