`3sqrt(512)div 4sqrt(16)div sqrt(576)=?`

Options

A. 24
B. 31
C. 22
D. None of these

Explanation

To solve the expression \( \frac{3\sqrt{512}}{4\sqrt{16}} \div \sqrt{576} \), we will break it down step by step.

### Step 1: Simplify \( \sqrt{512} \)
First, we need to simplify \( \sqrt{512} \).
\[ \sqrt{512} = \sqrt{256 \times 2} = \sqrt{256} \times \sqrt{2} = 16\sqrt{2} \]

### Step 2: Simplify \( \sqrt{16} \)
Next, we simplify \( \sqrt{16} \).
\[ \sqrt{16} = 4 \]

### Step 3: Simplify \( \sqrt{576} \)
Now, we simplify \( \sqrt{576} \).
\[ \sqrt{576} = 24 \]

### Step 4: Substitute the simplified values into the expression
Now we can substitute these values back into the original expression:
\[ \frac{3\sqrt{512}}{4\sqrt{16}} \div \sqrt{576} = \frac{3 \times 16\sqrt{2}}{4 \times 4} \div 24 \]

### Step 5: Calculate the numerator
Calculate the numerator:
\[ 3 \times 16\sqrt{2} = 48\sqrt{2} \]

### Step 6: Calculate the denominator
Calculate the denominator:
\[ 4 \times 4 = 16 \]

### Step 7: Combine the results
Now we can rewrite the expression:
\[ \frac{48\sqrt{2}}{16} \div 24 \]

### Step 8: Simplify the fraction
Simplifying \( \frac{48\sqrt{2}}{16} \):
\[ \frac{48}{16} = 3 \]
So we have:
\[ 3\sqrt{2} \div 24 \]

### Step 9: Final division
Now we perform the final division:
\[ \frac{3\sqrt{2}}{24} = \frac{\sqrt{2}}{8} \]

### Final Answer
Thus, the final answer is:
\[ \frac{\sqrt{2}}{8} \]

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