`(3^(2)-2^(2)) xx (frac(2)(3))^(-3)=` ?

Options

A. (a) `frac(45)(8)`
B. (b) `frac(8)(45)`
C. (c) `frac(8)(135)`
D. (d) `frac(135)(8)`

Answer

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

### Step 1: Calculate \(3^2\) and \(2^2\)
First, we calculate \(3^2\) and \(2^2\):
\[ 3^2 = 9 \]
\[ 2^2 = 4 \]

### Step 2: Subtract the results
Next, we subtract \(2^2\) from \(3^2\):
\[ 3^2 - 2^2 = 9 - 4 = 5 \]

### Step 3: Simplify \(\left(\frac{2}{3}\right)^{-3}\)
Now, we need to simplify \(\left(\frac{2}{3}\right)^{-3}\). A negative exponent means we take the reciprocal:
\[ \left(\frac{2}{3}\right)^{-3} = \left(\frac{3}{2}\right)^{3} \]

### Step 4: Calculate \(\left(\frac{3}{2}\right)^{3}\)
Now we calculate \(\left(\frac{3}{2}\right)^{3}\):
\[ \left(\frac{3}{2}\right)^{3} = \frac{3^3}{2^3} = \frac{27}{8} \]

### Step 5: Multiply the results
Finally, we multiply the results from Step 2 and Step 4:
\[ 5 \times \frac{27}{8} = \frac{5 \times 27}{8} = \frac{135}{8} \]

### Final Answer
Thus, the value of the expression \((3^2 - 2^2) \times \left(\frac{2}{3}\right)^{-3}\) is:
\[ \frac{135}{8} \]

---