`((16)/(81))^((-3)/(4))` is equal to

Options

A. `9/2`
B. `2/9`
C. `8/27`
D. `27/8`

Answer

To solve the expression \(\left(\frac{16}{81}\right)^{-\frac{3}{4}}\), we will follow these steps:

### Step 1: Factorize the numbers
First, we need to factorize 16 and 81.
- \(16 = 2^4\)
- \(81 = 3^4\)

### Step 2: Rewrite the expression
Now we can rewrite the expression using these factorizations:
\[ \left(\frac{16}{81}\right)^{-\frac{3}{4}} = \left(\frac{2^4}{3^4}\right)^{-\frac{3}{4}} \]

### Step 3: Apply the exponent property
Using the property of exponents \(\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}\), we can rewrite the expression as:
\[ \frac{(2^4)^{-\frac{3}{4}}}{(3^4)^{-\frac{3}{4}}} \]

### Step 4: Simplify the exponents
Now we simplify the exponents:
\[ (2^4)^{-\frac{3}{4}} = 2^{4 \cdot -\frac{3}{4}} = 2^{-3} \]
\[ (3^4)^{-\frac{3}{4}} = 3^{4 \cdot -\frac{3}{4}} = 3^{-3} \]

### Step 5: Rewrite the expression
Now we can rewrite the expression:
\[ \frac{2^{-3}}{3^{-3}} = \frac{1}{2^3} \div \frac{1}{3^3} = \frac{3^3}{2^3} \]

### Step 6: Calculate the final values
Now we calculate \(3^3\) and \(2^3\):
\[ 3^3 = 27 \quad \text{and} \quad 2^3 = 8 \]
Thus, we have:
\[ \frac{3^3}{2^3} = \frac{27}{8} \]

### Final Answer
The value of \(\left(\frac{16}{81}\right)^{-\frac{3}{4}}\) is \(\frac{27}{8}\).

---