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

Answer

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

### Step 1: Apply the exponent rule
Using the rule \(\left(\frac{A}{B}\right)^{M} = \frac{A^{M}}{B^{M}}\), we can rewrite the expression:

\[ \left(\frac{-3}{4}\right)^{3} = \frac{(-3)^{3}}{(4)^{3}}, \quad \left(\frac{-5}{2}\right)^{3} = \frac{(-5)^{3}}{(2)^{3}}, \quad \left(\frac{2}{3}\right)^{5} = \frac{(2)^{5}}{(3)^{5}} \]

So, the expression becomes:

\[ \frac{(-3)^{3}}{(4)^{3}} \times \frac{(-5)^{3}}{(2)^{3}} \times \frac{(2)^{5}}{(3)^{5}} \]

### Step 2: Calculate the powers
Now, we calculate the powers:

\[ (-3)^{3} = -27, \quad (4)^{3} = 64, \quad (-5)^{3} = -125, \quad (2)^{3} = 8, \quad (2)^{5} = 32, \quad (3)^{5} = 243 \]

Substituting these values back into the expression gives:

\[ \frac{-27}{64} \times \frac{-125}{8} \times \frac{32}{243} \]

### Step 3: Combine the fractions
Now we can combine the fractions:

\[ \frac{-27 \times -125 \times 32}{64 \times 8 \times 243} \]

### Step 4: Simplify the numerator and denominator
Calculating the numerator:

\[ -27 \times -125 = 3375 \quad \text{and} \quad 3375 \times 32 = 108000 \]

Calculating the denominator:

\[ 64 \times 8 = 512 \quad \text{and} \quad 512 \times 243 = 124416 \]

So we have:

\[ \frac{108000}{124416} \]

### Step 5: Simplify the fraction
Now we simplify \(\frac{108000}{124416}\). We can find the greatest common divisor (GCD) of 108000 and 124416, which is 144.

Dividing both the numerator and denominator by 144:

\[ \frac{108000 \div 144}{124416 \div 144} = \frac{750}{865} \]

### Step 6: Further simplify
Now, we can simplify \(\frac{750}{865}\) by dividing both by 5:

\[ \frac{150}{173} \]

Thus, the final simplified expression is:

\[ \frac{150}{173} \]

### Final Answer
The expression simplifies to \(\frac{150}{173}\), which is a rational number.

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