The value of `sqrt(9+2sqrt(16)+root3(512))` is :

Options

A. `6`
B. `5`
C. `2sqrt(8)`
D. `3sqrt(6)`

Answer

To solve the expression \( \sqrt{9 + 2\sqrt{16} + \sqrt[3]{512}} \), we will break it down step by step.

### Step 1: Simplify \( \sqrt{16} \)
The square root of 16 is:
\[ \sqrt{16} = 4 \]

### Step 2: Calculate \( 2\sqrt{16} \)
Now, we multiply the result from Step 1 by 2:
\[ 2\sqrt{16} = 2 \times 4 = 8 \]

### Step 3: Simplify \( \sqrt[3]{512} \)
Next, we find the cube root of 512. Since \( 512 = 8^3 \):
\[ \sqrt[3]{512} = 8 \]

### Step 4: Combine all parts
Now we can substitute the simplified values back into the original expression:
\[ 9 + 2\sqrt{16} + \sqrt[3]{512} = 9 + 8 + 8 \]

### Step 5: Calculate the sum
Now we add these values together:
\[ 9 + 8 + 8 = 25 \]

### Step 6: Find the square root
Finally, we take the square root of the sum:
\[ \sqrt{25} = 5 \]

Thus, the value of \( \sqrt{9 + 2\sqrt{16} + \sqrt[3]{512}} \) is \( 5 \).

### Final Answer
The answer is \( 5 \).

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