`((16)/(81))^(3//4) =`

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: Rewrite the numbers in terms of their prime factors.
- \(16\) can be expressed as \(2^4\) because \(16 = 2 \times 2 \times 2 \times 2\).
- \(81\) can be expressed as \(3^4\) because \(81 = 3 \times 3 \times 3 \times 3\).

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

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

### Step 3: Apply the power of a power rule.
Using the rule \((a^m)^n = a^{m \cdot n}\), we can simplify both the numerator and the denominator:
\[ \frac{2^{4 \cdot \frac{3}{4}}}{3^{4 \cdot \frac{3}{4}}} \]

Calculating the exponents:
- For the numerator: \(4 \cdot \frac{3}{4} = 3\)
- For the denominator: \(4 \cdot \frac{3}{4} = 3\)

So we have:
\[ \frac{2^3}{3^3} \]

### Step 4: Calculate the powers.
Now we can compute \(2^3\) and \(3^3\):
- \(2^3 = 2 \times 2 \times 2 = 8\)
- \(3^3 = 3 \times 3 \times 3 = 27\)

Thus, we have:
\[ \frac{8}{27} \]

### Final Answer:
Therefore, \(\left(\frac{16}{81}\right)^{\frac{3}{4}} = \frac{8}{27}\).

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