`(32)^((-2)/(5) ) div (125) ^((-2)/(3) )=`

Options

A. `4/25`
B. `25/4`
C. `2/5`
D. `5/2`

Answer

To solve the expression \((32)^{-\frac{2}{5}} \div (125)^{-\frac{2}{3}}\), we can follow these steps:

### Step 1: Rewrite the expression using positive exponents
Using the property of exponents that states \(a^{-b} = \frac{1}{a^b}\), we can rewrite the expression:

\[ (32)^{-\frac{2}{5}} = \frac{1}{(32)^{\frac{2}{5}}} \]
\[ (125)^{-\frac{2}{3}} = \frac{1}{(125)^{\frac{2}{3}}} \]

Thus, the expression becomes:

\[ \frac{1}{(32)^{\frac{2}{5}}} \div \frac{1}{(125)^{\frac{2}{3}}} \]

### Step 2: Change division to multiplication
Dividing by a fraction is equivalent to multiplying by its reciprocal. Therefore, we can rewrite the expression as:

\[ \frac{1}{(32)^{\frac{2}{5}}} \times (125)^{\frac{2}{3}} \]

### Step 3: Combine the expression
This can be rewritten as:

\[ \frac{(125)^{\frac{2}{3}}}{(32)^{\frac{2}{5}}} \]

### Step 4: Simplify the bases
Next, we can express \(32\) and \(125\) in terms of their prime factors:

\[ 32 = 2^5 \quad \text{and} \quad 125 = 5^3 \]

Now substituting these values into the expression gives:

\[ \frac{(5^3)^{\frac{2}{3}}}{(2^5)^{\frac{2}{5}}} \]

### Step 5: Apply the power of a power property
Using the property \((a^m)^n = a^{m \cdot n}\), we can simplify:

\[ (5^3)^{\frac{2}{3}} = 5^{3 \cdot \frac{2}{3}} = 5^2 \]
\[ (2^5)^{\frac{2}{5}} = 2^{5 \cdot \frac{2}{5}} = 2^2 \]

Thus, the expression simplifies to:

\[ \frac{5^2}{2^2} \]

### Step 6: Calculate the final values
Now, we can calculate \(5^2\) and \(2^2\):

\[ 5^2 = 25 \quad \text{and} \quad 2^2 = 4 \]

So the expression becomes:

\[ \frac{25}{4} \]

### Final Answer
The final answer is:

\[ \frac{25}{4} \]

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