`{(frac(1)(3))^(2)}^(4)=` ?

Options

A. (a) `(frac(1)(3))^(6)`
B. (b) `(frac(1)(3))^(8)`
C. (c) `(frac(1)(3))^(16)`
D. (d) `(frac(1)(3))^(24)`

Answer

To solve the expression \(\left(\frac{1}{3}\right)^{2}\) raised to the power of 4, we can follow these steps:

### Step 1: Understand the expression
The expression we have is \(\left(\frac{1}{3}\right)^{2}\) raised to the power of 4. This can be written as:
\[ \left(\left(\frac{1}{3}\right)^{2}\right)^{4} \]

### Step 2: Apply the power of a power rule
According to the power of a power rule in exponents, when you raise a power to another power, you multiply the exponents. Therefore, we can rewrite the expression as:
\[ \left(\frac{1}{3}\right)^{2 \times 4} \]

### Step 3: Calculate the new exponent
Now, calculate the multiplication of the exponents:
\[ 2 \times 4 = 8 \]
So, we have:
\[ \left(\frac{1}{3}\right)^{8} \]

### Step 4: Write the final answer
Thus, the final answer is:
\[ \left(\frac{1}{3}\right)^{8} \]

### Summary of the solution:
The expression \(\left(\frac{1}{3}\right)^{2}\) raised to the power of 4 simplifies to \(\left(\frac{1}{3}\right)^{8}\).

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