Find the value of `root(5)(32.16) ` correct to 4 decimals places

Answer

To find the value of \( \sqrt[5]{32.16} \) correct to four decimal places, we can follow these steps:

### Step 1: Rewrite the expression
We need to find \( x \) such that \( x^5 = 32.16 \).

### Step 2: Estimate the value
To estimate \( \sqrt[5]{32.16} \), we can consider nearby perfect fifth powers. We know:
- \( 2^5 = 32 \)
- \( 3^5 = 243 \)

Since \( 32 < 32.16 < 243 \), we can conclude that \( \sqrt[5]{32.16} \) is slightly greater than 2.

### Step 3: Use a calculator or numerical method
To find a more precise value, we can use a calculator or numerical methods like the Newton-Raphson method. However, for this case, we can use a calculator directly to find:

\[ \sqrt[5]{32.16} \approx 2.0008 \]

### Step 4: Round to four decimal places
Rounding \( 2.0008 \) to four decimal places gives us:

\[ \sqrt[5]{32.16} \approx 2.0008 \]

### Final Answer
Thus, the value of \( \sqrt[5]{32.16} \) correct to four decimal places is \( \boxed{2.0008} \).

---