When `((81)/(16))^((-3)/(4)) xx {((9)/(25))^((5)/(2)) div ( (5)/(2))^(-3)}` is simplified , we get .

Options

A. `(125)/(27)`
B. `(27)/(125)`
C. 1
D. `(9)/(25)`

Answer

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

### Step 1: Rewrite the bases in terms of prime factors
We can express \(81\), \(16\), \(9\), and \(25\) as powers of their prime factors:
- \(81 = 3^4\)
- \(16 = 2^4\)
- \(9 = 3^2\)
- \(25 = 5^2\)

Thus, we can rewrite the expression as:
\[ \left(\frac{3^4}{2^4}\right)^{-\frac{3}{4}} \times \left(\frac{3^2}{5^2}\right)^{\frac{5}{2}} \div \left(\frac{5}{2}\right)^{-3} \]

### Step 2: Apply the power of a quotient rule
Using the rule \((\frac{a}{b})^n = \frac{a^n}{b^n}\), we can simplify:
\[ \left(\frac{3^4}{2^4}\right)^{-\frac{3}{4}} = \frac{(3^4)^{-\frac{3}{4}}}{(2^4)^{-\frac{3}{4}}} = \frac{3^{-3}}{2^{-3}} = \frac{2^3}{3^3} \]

### Step 3: Simplify the second term
Now simplify the second term:
\[ \left(\frac{3^2}{5^2}\right)^{\frac{5}{2}} = \frac{(3^2)^{\frac{5}{2}}}{(5^2)^{\frac{5}{2}}} = \frac{3^5}{5^5} \]

### Step 4: Rewrite the division
The division can be rewritten as multiplication by the reciprocal:
\[ \left(\frac{5}{2}\right)^{-3} = \frac{2^3}{5^3} \]

### Step 5: Combine all parts
Now we can combine everything:
\[ \frac{2^3}{3^3} \times \frac{3^5}{5^5} \times \frac{2^3}{5^3} \]

### Step 6: Combine the fractions
Combine the numerators and denominators:
\[ \frac{2^3 \times 3^5 \times 2^3}{3^3 \times 5^5 \times 5^3} = \frac{2^{3+3} \times 3^5}{3^3 \times 5^{5+3}} = \frac{2^6 \times 3^5}{3^3 \times 5^8} \]

### Step 7: Simplify the powers
Now simplify the powers of \(3\):
\[ \frac{2^6 \times 3^{5-3}}{5^8} = \frac{2^6 \times 3^2}{5^8} \]

### Step 8: Calculate the final result
Now calculate \(3^2\) and \(2^6\):
\[ 3^2 = 9 \quad \text{and} \quad 2^6 = 64 \]
Thus, we have:
\[ \frac{64 \times 9}{5^8} = \frac{576}{625} \]

### Final Answer
The simplified expression is:
\[ \frac{576}{625} \]