Simplify : `((25)/(9))^(-(3)/(2))xx((2)/(5))^(-3)`

Answer

To simplify the expression \(\left(\frac{25}{9}\right)^{-\frac{3}{2}} \times \left(\frac{2}{5}\right)^{-3}\), we can follow these steps:

### Step 1: Rewrite the bases
We can express \(25\) and \(9\) in terms of their prime factors:
\[ 25 = 5^2 \quad \text{and} \quad 9 = 3^2 \]
Thus, we can rewrite the expression as:
\[ \left(\frac{5^2}{3^2}\right)^{-\frac{3}{2}} \times \left(\frac{2}{5}\right)^{-3} \]

### Step 2: Apply the negative exponent rule
Using the property \(a^{-m} = \frac{1}{a^m}\), we can simplify the first part:
\[ \left(\frac{5^2}{3^2}\right)^{-\frac{3}{2}} = \left(\frac{3^2}{5^2}\right)^{\frac{3}{2}} \]
Now the expression becomes:
\[ \left(\frac{3^2}{5^2}\right)^{\frac{3}{2}} \times \left(\frac{2}{5}\right)^{-3} \]

### Step 3: Simplify the first part
Using the property \((\frac{a}{b})^m = \frac{a^m}{b^m}\):
\[ \left(\frac{3^2}{5^2}\right)^{\frac{3}{2}} = \frac{3^{2 \cdot \frac{3}{2}}}{5^{2 \cdot \frac{3}{2}}} = \frac{3^3}{5^3} \]
So now we have:
\[ \frac{3^3}{5^3} \times \left(\frac{2}{5}\right)^{-3} \]

### Step 4: Simplify the second part
Using the negative exponent rule again:
\[ \left(\frac{2}{5}\right)^{-3} = \frac{5^3}{2^3} \]
Now the expression becomes:
\[ \frac{3^3}{5^3} \times \frac{5^3}{2^3} \]

### Step 5: Combine the fractions
Now we can multiply the fractions:
\[ \frac{3^3 \times 5^3}{5^3 \times 2^3} \]
The \(5^3\) in the numerator and denominator cancels out:
\[ \frac{3^3}{2^3} \]

### Step 6: Final simplification
Now we can simplify this to:
\[ \frac{27}{8} \]

Thus, the final simplified expression is:
\[ \frac{27}{8} \]