Simplify :
`((81)/(16))^(-(3)/(4))xx{((25)/(9))^(-(3)/(2))/((5)/(2))^(-3)}`

Options

A. `1`
B. `2`
C. `0`
D. `4`

Answer

To simplify the expression \(\left(\frac{81}{16}\right)^{-\frac{3}{4}} \times \frac{\left(\frac{25}{9}\right)^{-\frac{3}{2}}}{\left(\frac{5}{2}\right)^{-3}}\), we will follow these steps:

### Step 1: Apply the Negative Exponent Rule
Using the rule \(a^{-b} = \frac{1}{a^b}\), we can rewrite the expression:
\[ \left(\frac{81}{16}\right)^{-\frac{3}{4}} = \frac{1}{\left(\frac{81}{16}\right)^{\frac{3}{4}}} \]
\[ \left(\frac{25}{9}\right)^{-\frac{3}{2}} = \frac{1}{\left(\frac{25}{9}\right)^{\frac{3}{2}}} \]
\[ \left(\frac{5}{2}\right)^{-3} = \frac{1}{\left(\frac{5}{2}\right)^{3}} \]
Thus, the expression becomes:
\[ \frac{1}{\left(\frac{81}{16}\right)^{\frac{3}{4}}} \times \frac{\frac{1}{\left(\frac{25}{9}\right)^{\frac{3}{2}}}}{\frac{1}{\left(\frac{5}{2}\right)^{3}}} \]

### Step 2: Simplify the Fraction
This simplifies to:
\[ \frac{\left(\frac{5}{2}\right)^{3}}{\left(\frac{81}{16}\right)^{\frac{3}{4}} \times \left(\frac{25}{9}\right)^{\frac{3}{2}}} \]

### Step 3: Calculate Each Component
1. **Calculate \(\left(\frac{81}{16}\right)^{\frac{3}{4}}\)**:
\[ \left(\frac{81}{16}\right)^{\frac{3}{4}} = \frac{81^{\frac{3}{4}}}{16^{\frac{3}{4}}} \]
- \(81 = 3^4\) so \(81^{\frac{3}{4}} = (3^4)^{\frac{3}{4}} = 3^3 = 27\)
- \(16 = 2^4\) so \(16^{\frac{3}{4}} = (2^4)^{\frac{3}{4}} = 2^3 = 8\)
- Thus, \(\left(\frac{81}{16}\right)^{\frac{3}{4}} = \frac{27}{8}\)

2. **Calculate \(\left(\frac{25}{9}\right)^{\frac{3}{2}}\)**:
\[ \left(\frac{25}{9}\right)^{\frac{3}{2}} = \frac{25^{\frac{3}{2}}}{9^{\frac{3}{2}}} \]
- \(25 = 5^2\) so \(25^{\frac{3}{2}} = (5^2)^{\frac{3}{2}} = 5^3 = 125\)
- \(9 = 3^2\) so \(9^{\frac{3}{2}} = (3^2)^{\frac{3}{2}} = 3^3 = 27\)
- Thus, \(\left(\frac{25}{9}\right)^{\frac{3}{2}} = \frac{125}{27}\)

### Step 4: Substitute Back into the Expression
Now substituting these values back, we have:
\[ \frac{\left(\frac{5}{2}\right)^{3}}{\frac{27}{8} \times \frac{125}{27}} \]

### Step 5: Simplify the Denominator
The denominator simplifies as follows:
\[ \frac{27}{8} \times \frac{125}{27} = \frac{125}{8} \]
Thus, we have:
\[ \frac{\left(\frac{5}{2}\right)^{3}}{\frac{125}{8}} = \left(\frac{5}{2}\right)^{3} \times \frac{8}{125} \]

### Step 6: Calculate \(\left(\frac{5}{2}\right)^{3}\)
Calculating \(\left(\frac{5}{2}\right)^{3}\):
\[ \left(\frac{5}{2}\right)^{3} = \frac{125}{8} \]

### Step 7: Final Simplification
Now substituting this back:
\[ \frac{125}{8} \times \frac{8}{125} = 1 \]

### Conclusion
Thus, the simplified expression is:
\[ \boxed{1} \]