Evaluate `(0.00032)^(2//5)`

Options

A. `1/625`
B. `1/225`
C. `1/125`
D. `1/25`

Answer

To evaluate \( (0.00032)^{\frac{2}{5}} \), we can follow these steps:

### Step 1: Convert the decimal to a fraction
First, we convert \( 0.00032 \) into a fraction.

\[ 0.00032 = \frac{32}{10000} \]

### Step 2: Simplify the fraction
Next, we simplify \( \frac{32}{10000} \).

The prime factorization of \( 32 \) is:
\[ 32 = 2^5 \]

The prime factorization of \( 10000 \) is:
\[ 10000 = 10^4 = (2 \times 5)^4 = 2^4 \times 5^4 \]

Thus, we can write:
\[ \frac{32}{10000} = \frac{2^5}{2^4 \times 5^4} \]

### Step 3: Cancel common factors
Now we can simplify the fraction:
\[ \frac{2^5}{2^4 \times 5^4} = \frac{2^{5-4}}{5^4} = \frac{2^1}{5^4} = \frac{2}{625} \]

### Step 4: Apply the exponent
Now we need to raise this fraction to the power of \( \frac{2}{5} \):
\[ \left(\frac{2}{625}\right)^{\frac{2}{5}} \]

Using the property of exponents, we can distribute the exponent:
\[ \frac{2^{\frac{2}{5}}}{625^{\frac{2}{5}}} \]

### Step 5: Simplify the denominator
Next, we simplify \( 625^{\frac{2}{5}} \). Since \( 625 = 5^4 \):
\[ 625^{\frac{2}{5}} = (5^4)^{\frac{2}{5}} = 5^{\frac{4 \times 2}{5}} = 5^{\frac{8}{5}} = 5^{1.6} \]

### Step 6: Combine the results
Now we can write our final expression:
\[ \frac{2^{\frac{2}{5}}}{5^{1.6}} \]

### Step 7: Calculate the values
Calculating \( 2^{\frac{2}{5}} \) and \( 5^{1.6} \) can be done using a calculator or logarithm tables, but we can also leave it in this form for an exact answer.

### Final Answer
Thus, the evaluated result of \( (0.00032)^{\frac{2}{5}} \) is:
\[ \frac{2^{\frac{2}{5}}}{5^{1.6}} \]