Simplify and express each of the following as a rational number :
` ((6)/(5))^(3) xx ( ( 5)/(2))^(2) `

Answer

To simplify the expression \(\left(\frac{6}{5}\right)^{3} \times \left(\frac{5}{2}\right)^{2}\) and express it as a rational number, follow these steps:

### Step 1: Rewrite the expression
Start with the original expression:
\[ \left(\frac{6}{5}\right)^{3} \times \left(\frac{5}{2}\right)^{2} \]

### Step 2: Expand the fractions
Rewrite \(6\) as \(2 \times 3\):
\[ \left(\frac{2 \times 3}{5}\right)^{3} \times \left(\frac{5}{2}\right)^{2} \]

### Step 3: Apply the exponent to both the numerator and the denominator
This gives us:
\[ \frac{(2 \times 3)^{3}}{5^{3}} \times \frac{5^{2}}{2^{2}} \]

### Step 4: Simplify the expression
Now, simplify each part:
\[ \frac{2^{3} \times 3^{3}}{5^{3}} \times \frac{5^{2}}{2^{2}} \]

### Step 5: Combine the fractions
Combine the fractions:
\[ \frac{2^{3} \times 3^{3} \times 5^{2}}{5^{3} \times 2^{2}} \]

### Step 6: Simplify using the laws of exponents
Now, apply the laws of exponents:
\[ \frac{2^{3-2} \times 3^{3}}{5^{3-2}} = \frac{2^{1} \times 3^{3}}{5^{1}} = \frac{2 \times 27}{5} \]

### Step 7: Calculate the final result
Now, calculate:
\[ \frac{54}{5} \]

Thus, the simplified expression is:
\[ \frac{54}{5} \]