Simplify and express each of the following as a rational number .
` ((3)/(4))^(2) xx ((-2)/( 5))^(3) xx 4^(3)`

Answer

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

### Step 1: Rewrite the expression using the properties of exponents
We can express the powers of fractions as the powers of their numerators and denominators:
\[ \left(\frac{3}{4}\right)^{2} = \frac{3^{2}}{4^{2}} \quad \text{and} \quad \left(-\frac{2}{5}\right)^{3} = \frac{-2^{3}}{5^{3}} \]
Thus, we can rewrite the expression as:
\[ \frac{3^{2}}{4^{2}} \times \frac{-2^{3}}{5^{3}} \times 4^{3} \]

### Step 2: Combine the fractions
Now we can combine the fractions:
\[ \frac{3^{2} \times (-2^{3}) \times 4^{3}}{4^{2} \times 5^{3}} \]

### Step 3: Simplify the expression
We can simplify \(4^{3}\) and \(4^{2}\):
\[ \frac{3^{2} \times (-2^{3}) \times 4^{3}}{4^{2} \times 5^{3}} = \frac{3^{2} \times (-2^{3}) \times 4^{3-2}}{5^{3}} = \frac{3^{2} \times (-2^{3}) \times 4^{1}}{5^{3}} \]

### Step 4: Calculate the individual powers
Now we calculate the individual powers:
- \(3^{2} = 9\)
- \(-2^{3} = -8\)
- \(4^{1} = 4\)

Substituting these values back into the expression gives:
\[ \frac{9 \times (-8) \times 4}{5^{3}} \]

### Step 5: Calculate \(5^{3}\)
Calculate \(5^{3}\):
\[ 5^{3} = 125 \]

### Step 6: Multiply the numbers in the numerator
Now, we multiply the numbers in the numerator:
\[ 9 \times (-8) = -72 \]
Then multiply by \(4\):
\[ -72 \times 4 = -288 \]

### Step 7: Write the final expression
Now we can write the final expression:
\[ \frac{-288}{125} \]

Thus, the simplified expression is:
\[ \frac{-288}{125} \]