`root(6)(0.004096)=`

Options

A. 0.2
B. 0.4
C. 0.6
D. 0.8

Explanation

To solve the expression \( \sqrt[6]{0.004096} \), we can follow these steps:

### Step 1: Rewrite the number in scientific notation
First, we can express \( 0.004096 \) in a more manageable form. We can write it as:
\[ 0.004096 = \frac{4096}{1000} = \frac{4096}{10^3} \]

### Step 2: Rewrite the expression
Now we can rewrite the original expression using this fraction:
\[ \sqrt[6]{0.004096} = \sqrt[6]{\frac{4096}{10^3}} = \frac{\sqrt[6]{4096}}{\sqrt[6]{10^3}} \]

### Step 3: Calculate \( \sqrt[6]{4096} \)
Next, we need to find \( \sqrt[6]{4096} \). We can factor \( 4096 \) to find its prime factors:
\[ 4096 = 2^{12} \quad \text{(since \( 4096 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \))} \]
Now, we can find the sixth root:
\[ \sqrt[6]{4096} = \sqrt[6]{2^{12}} = 2^{12/6} = 2^2 = 4 \]

### Step 4: Calculate \( \sqrt[6]{10^3} \)
Now we calculate \( \sqrt[6]{10^3} \):
\[ \sqrt[6]{10^3} = 10^{3/6} = 10^{1/2} = \sqrt{10} \]

### Step 5: Combine the results
Now we can combine our results:
\[ \sqrt[6]{0.004096} = \frac{\sqrt[6]{4096}}{\sqrt[6]{10^3}} = \frac{4}{\sqrt{10}} \]

### Step 6: Rationalize the denominator
To express the answer in a more standard form, we can rationalize the denominator:
\[ \frac{4}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}} = \frac{4\sqrt{10}}{10} = \frac{2\sqrt{10}}{5} \]

### Final Answer
Thus, the final answer is:
\[ \sqrt[6]{0.004096} = \frac{2\sqrt{10}}{5} \]

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