the value of ` sqrt(root (3) (0.000729 ))` is

Options

A. `0.03`
B. `0.09`
C. `0.9`
D. `0.3`

Answer

To solve the problem of finding the value of \( \sqrt[2]{\sqrt[3]{0.000729}} \), we will break it down into steps.

### Step 1: Convert the decimal to a fraction
First, we convert \( 0.000729 \) into a fraction.
\[ 0.000729 = \frac{729}{1000000} \]

### Step 2: Find the cube root
Next, we need to find the cube root of \( 0.000729 \) or \( \frac{729}{1000000} \).
\[ \sqrt[3]{0.000729} = \sqrt[3]{\frac{729}{1000000}} = \frac{\sqrt[3]{729}}{\sqrt[3]{1000000}} \]
We know that \( 729 = 9^3 \), so \( \sqrt[3]{729} = 9 \).
Also, \( 1000000 = 10^6 \), so \( \sqrt[3]{1000000} = 10^2 = 100 \).
Thus,
\[ \sqrt[3]{0.000729} = \frac{9}{100} \]

### Step 3: Find the square root
Now we will find the square root of \( \frac{9}{100} \).
\[ \sqrt{\frac{9}{100}} = \frac{\sqrt{9}}{\sqrt{100}} = \frac{3}{10} \]

### Step 4: Convert to decimal (if needed)
The fraction \( \frac{3}{10} \) can also be expressed as a decimal:
\[ \frac{3}{10} = 0.3 \]

### Final Answer
Thus, the value of \( \sqrt[2]{\sqrt[3]{0.000729}} \) is \( 0.3 \).

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