Evaluate `((3)/(8))^(-2) xx ((4)/(5))^(-3).`

Options

A. `(121)/(9)`
B. `(125)/(9)`
C. `(155)/(9)`
D. `(15)/(9)`

Answer

To evaluate the expression \(\left(\frac{3}{8}\right)^{-2} \times \left(\frac{4}{5}\right)^{-3}\), we will follow these steps:

### Step 1: Apply the Negative Exponent Rule
The negative exponent rule states that \(a^{-m} = \frac{1}{a^m}\). Therefore, we can rewrite the expression:

\[ \left(\frac{3}{8}\right)^{-2} = \frac{1}{\left(\frac{3}{8}\right)^{2}} \quad \text{and} \quad \left(\frac{4}{5}\right)^{-3} = \frac{1}{\left(\frac{4}{5}\right)^{3}} \]

So, we can rewrite the original expression as:

\[ \frac{1}{\left(\frac{3}{8}\right)^{2}} \times \frac{1}{\left(\frac{4}{5}\right)^{3}} = \frac{1}{\left(\frac{3}{8}\right)^{2} \times \left(\frac{4}{5}\right)^{3}} \]

### Step 2: Calculate the Powers
Now we will calculate \(\left(\frac{3}{8}\right)^{2}\) and \(\left(\frac{4}{5}\right)^{3}\):

\[ \left(\frac{3}{8}\right)^{2} = \frac{3^{2}}{8^{2}} = \frac{9}{64} \]

\[ \left(\frac{4}{5}\right)^{3} = \frac{4^{3}}{5^{3}} = \frac{64}{125} \]

### Step 3: Multiply the Results
Now we will multiply these two results:

\[ \left(\frac{3}{8}\right)^{2} \times \left(\frac{4}{5}\right)^{3} = \frac{9}{64} \times \frac{64}{125} \]

### Step 4: Simplify the Expression
When multiplying fractions, we multiply the numerators and the denominators:

\[ \frac{9 \times 64}{64 \times 125} = \frac{9}{125} \]

### Step 5: Write the Final Result
Now, substituting back into our expression:

\[ \frac{1}{\left(\frac{3}{8}\right)^{2} \times \left(\frac{4}{5}\right)^{3}} = \frac{1}{\frac{9}{125}} = \frac{125}{9} \]

Thus, the final answer is:

\[ \frac{125}{9} \]

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