Value of ? In `root(3)(512)+root(4)(16)+sqrt576=?` is

Options

A. 24
B. 31
C. 22
D. None of the above

Answer

To find the value of the expression \( \sqrt[3]{512} + \sqrt[4]{16} + \sqrt{576} \), we will break it down step by step.

### Step 1: Calculate \( \sqrt[3]{512} \)
First, we need to find the cube root of 512.

- **Prime Factorization of 512**:
\[ 512 = 2^9 \quad (\text{since } 512 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2) \]
- **Cube Root Calculation**:
\[ \sqrt[3]{512} = \sqrt[3]{2^9} = 2^{9/3} = 2^3 = 8 \]

### Step 2: Calculate \( \sqrt[4]{16} \)
Next, we find the fourth root of 16.

- **Prime Factorization of 16**:
\[ 16 = 2^4 \]
- **Fourth Root Calculation**:
\[ \sqrt[4]{16} = \sqrt[4]{2^4} = 2^{4/4} = 2^1 = 2 \]

### Step 3: Calculate \( \sqrt{576} \)
Now, we will calculate the square root of 576.

- **Prime Factorization of 576**:
\[ 576 = 24 \times 24 = (2^3 \times 3)^2 = 2^6 \times 3^2 \]
- **Square Root Calculation**:
\[ \sqrt{576} = \sqrt{2^6 \times 3^2} = \sqrt{2^6} \times \sqrt{3^2} = 2^{6/2} \times 3^{2/2} = 2^3 \times 3^1 = 8 \times 3 = 24 \]

### Step 4: Combine the Results
Now we add the results from the previous steps:
\[ \sqrt[3]{512} + \sqrt[4]{16} + \sqrt{576} = 8 + 2 + 24 \]
Calculating this gives:
\[ 8 + 2 + 24 = 34 \]

### Final Answer
Thus, the value of the expression is:
\[ \boxed{34} \]