`sqrt(3sqrt(0.004096))` is equal to

Options

A. 4
B. 0.4
C. 0.04
D. 0.004

Answer

To solve the expression \( \sqrt{3\sqrt{0.004096}} \), we will follow these steps:

### Step 1: Simplify the inner square root
First, we need to find the value of \( \sqrt{0.004096} \).

To do this, we can express \( 0.004096 \) in a more manageable form. Notice that:
\[ 0.004096 = \frac{4096}{1000000} = \frac{4096}{10^6} \]

### Step 2: Calculate the square root of 4096
Next, we need to find \( \sqrt{4096} \). We know that:
\[ 4096 = 64^2 \quad \text{(since \( 64 \times 64 = 4096 \))} \]
Thus,
\[ \sqrt{4096} = 64 \]

### Step 3: Calculate the square root of \( 10^6 \)
Now, we calculate \( \sqrt{10^6} \):
\[ \sqrt{10^6} = 10^{6/2} = 10^3 = 1000 \]

### Step 4: Combine the results
Now we combine the results from Steps 2 and 3:
\[ \sqrt{0.004096} = \frac{\sqrt{4096}}{\sqrt{10^6}} = \frac{64}{1000} = 0.064 \]

### Step 5: Multiply by 3
Now we substitute back into the original expression:
\[ 3\sqrt{0.004096} = 3 \times 0.064 = 0.192 \]

### Step 6: Take the square root of the result
Finally, we need to find \( \sqrt{0.192} \). To simplify this, we can express \( 0.192 \) as:
\[ 0.192 = \frac{192}{1000} = \frac{192}{10^3} \]

Calculating \( \sqrt{192} \):
\[ 192 = 64 \times 3 = 8^2 \times 3 \quad \text{(since \( 8 \times 8 = 64 \))} \]
Thus,
\[ \sqrt{192} = 8\sqrt{3} \]

And for \( \sqrt{10^3} \):
\[ \sqrt{10^3} = 10^{3/2} = 10 \sqrt{10} \]

### Step 7: Combine the square roots
Now we can combine:
\[ \sqrt{0.192} = \frac{\sqrt{192}}{\sqrt{10^3}} = \frac{8\sqrt{3}}{10\sqrt{10}} = \frac{8\sqrt{3}}{10\sqrt{10}} = \frac{4\sqrt{3}}{5\sqrt{10}} \]

### Final Answer
Thus, the final answer for \( \sqrt{3\sqrt{0.004096}} \) is:
\[ \frac{4\sqrt{3}}{5\sqrt{10}} \]