If `(2^(3))^(2)=4^(x)` then `3^(x)` is equal to

Options

A. 3
B. 6
C. 9
D. 27

Answer

To solve the equation \( (2^3)^2 = 4^x \) and find the value of \( 3^x \), we can follow these steps:

### Step 1: Simplify the Left Side
We start with the left side of the equation:
\[ (2^3)^2 \]
Using the power of a power property, which states that \( (a^m)^n = a^{m \cdot n} \), we can simplify this:
\[ (2^3)^2 = 2^{3 \cdot 2} = 2^6 \]

### Step 2: Rewrite the Right Side
Next, we rewrite the right side of the equation:
\[ 4^x \]
We know that \( 4 \) can be expressed as \( 2^2 \), so we rewrite it:
\[ 4^x = (2^2)^x \]
Again, applying the power of a power property:
\[ (2^2)^x = 2^{2x} \]

### Step 3: Set the Exponents Equal
Now we have:
\[ 2^6 = 2^{2x} \]
Since the bases are the same, we can set the exponents equal to each other:
\[ 6 = 2x \]

### Step 4: Solve for \( x \)
To find \( x \), we divide both sides by 2:
\[ x = \frac{6}{2} = 3 \]

### Step 5: Find \( 3^x \)
Now that we have \( x = 3 \), we need to find \( 3^x \):
\[ 3^x = 3^3 \]
Calculating \( 3^3 \):
\[ 3^3 = 3 \times 3 \times 3 = 27 \]

### Final Answer
Thus, the value of \( 3^x \) is:
\[ \boxed{27} \]