Find the value of `sqrt(sqrt(sqrt(0.00000256)))`

Options

A. 0.4
B. 0.02
C. 0.04
D. 0.2

Answer

To find the value of \( \sqrt{\sqrt{\sqrt{0.00000256}}} \), we can follow these steps:

### Step 1: Rewrite the number in scientific notation
First, we can express \( 0.00000256 \) in scientific notation.
\[ 0.00000256 = 2.56 \times 10^{-6} \]

### Step 2: Take the first square root
Now, we will take the square root of \( 2.56 \times 10^{-6} \).
\[ \sqrt{2.56 \times 10^{-6}} = \sqrt{2.56} \times \sqrt{10^{-6}} \]
Calculating each part:
- \( \sqrt{2.56} = 1.6 \)
- \( \sqrt{10^{-6}} = 10^{-3} \)

So,
\[ \sqrt{2.56 \times 10^{-6}} = 1.6 \times 10^{-3} \]

### Step 3: Take the second square root
Next, we take the square root of \( 1.6 \times 10^{-3} \).
\[ \sqrt{1.6 \times 10^{-3}} = \sqrt{1.6} \times \sqrt{10^{-3}} \]
Calculating each part:
- \( \sqrt{1.6} \approx 1.2649 \)
- \( \sqrt{10^{-3}} = 10^{-1.5} = 10^{-1} \times 10^{-0.5} = 0.1 \times \sqrt{10} \approx 0.3162 \)

So,
\[ \sqrt{1.6 \times 10^{-3}} \approx 1.2649 \times 0.3162 \approx 0.4 \]

### Step 4: Take the third square root
Finally, we take the square root of \( 0.4 \).
\[ \sqrt{0.4} = \sqrt{\frac{4}{10}} = \frac{2}{\sqrt{10}} \approx \frac{2}{3.1623} \approx 0.6325 \]

### Final Result
Thus, the value of \( \sqrt{\sqrt{\sqrt{0.00000256}}} \) is approximately \( 0.6325 \).