`root(4)((0.00002025)/(0.00005329))` is equal to

Options

A. `sqrt(45/73)`
B. `sqrt(43/75)`
C. `sqrt(34/57)`
D. `sqrt(73/45)`

Answer

To solve the expression \( \sqrt[4]{\frac{0.00002025}{0.00005329}} \), we can follow these steps:

### Step 1: Simplify the fraction
First, we can rewrite the fraction in a more manageable form by removing the decimal points.

\[ \frac{0.00002025}{0.00005329} = \frac{2025}{5329} \]

### Step 2: Factor the numerator and denominator
Next, we need to factor both the numerator and the denominator.

The numerator \( 2025 \) can be factored as follows:
\[ 2025 = 45^2 = (3^2 \cdot 5)^2 = 3^4 \cdot 5^2 \]

The denominator \( 5329 \) can be factored as:
\[ 5329 = 73^2 \]

### Step 3: Rewrite the expression
Now we can rewrite the expression using these factors:
\[ \frac{2025}{5329} = \frac{3^4 \cdot 5^2}{73^2} \]

### Step 4: Apply the fourth root
Now we apply the fourth root to the entire fraction:
\[ \sqrt[4]{\frac{3^4 \cdot 5^2}{73^2}} = \frac{\sqrt[4]{3^4} \cdot \sqrt[4]{5^2}}{\sqrt[4]{73^2}} \]

### Step 5: Simplify the roots
Now we simplify each part:
\[ \sqrt[4]{3^4} = 3, \quad \sqrt[4]{5^2} = 5^{1/2} = \sqrt{5}, \quad \sqrt[4]{73^2} = 73^{1/2} = \sqrt{73} \]

Putting it all together, we have:
\[ \frac{3 \cdot \sqrt{5}}{\sqrt{73}} \]

### Step 6: Final expression
Thus, the final expression is:
\[ \sqrt[4]{\frac{0.00002025}{0.00005329}} = \frac{3 \sqrt{5}}{\sqrt{73}} \]