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

Answer

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

### Step 1: Apply the Exponent Rule
Using the exponent rule \(\left(\frac{a}{b}\right)^{m} = \frac{a^{m}}{b^{m}}\), we can rewrite the expression:

\[ \left(\frac{5}{4}\right)^{2} = \frac{5^{2}}{4^{2}} = \frac{25}{16} \]
\[ \left(-\frac{3}{5}\right)^{3} = \frac{(-3)^{3}}{5^{3}} = \frac{-27}{125} \]

Now, substituting these back into the expression gives us:

\[ \frac{25}{16} \times \frac{2}{3} \times \frac{-27}{125} \]

### Step 2: Multiply the Numerators and Denominators
Now we multiply the numerators and the denominators:

Numerator:
\[ 25 \times 2 \times (-27) = 50 \times (-27) = -1350 \]

Denominator:
\[ 16 \times 3 \times 125 = 48 \times 125 = 6000 \]

So we have:

\[ \frac{-1350}{6000} \]

### Step 3: Simplify the Fraction
Now we need to simplify \(\frac{-1350}{6000}\). We can find the greatest common divisor (GCD) of 1350 and 6000.

The GCD of 1350 and 6000 is 150. Now we divide both the numerator and the denominator by 150:

\[ \frac{-1350 \div 150}{6000 \div 150} = \frac{-9}{40} \]

### Final Answer
Thus, the simplified expression is:

\[ \frac{-9}{40} \]

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