Solve ((a×3)²)⁴: Evaluating Nested Exponent Expression

To solve this problem, we'll apply the power of a power rule from exponents:

• Step 1: Identify the base and the exponents involved.

• Step 2: Apply the power of a power rule.

• Step 3: Simplify the expression by multiplying the exponents.

Now, let's work through each step:
Step 1: The original expression is $\left(\left(a\times3\right)^2\right)^4$. We recognize the base as $a \times 3$ and see it is first raised to the power of 2, and then the result is raised to the power of 4.
Step 2: We'll use the power of a power property of exponents: $(b^m)^n = b^{m \times n}$. Here, $b$ can be considered as $(a \times 3)$, $m = 2$, and $n = 4$.
Step 3: Applying this property, we have $\left(\left(a\times3\right)^2\right)^4 = \left(a\times3\right)^{2 \times 4}$.
By multiplying the exponents, we get $(a \times 3)^8$, but to match the format requested in the choices, we simply express it as $(a \times 3)^{2 \times 4}$.

Therefore, the correct expression that corresponds to the given power structure is $\left(a\times3\right)^{2\times4}$.

Analyzing the choices provided:

• Choice 1: $\left(a\times3\right)^{4-2}$ applies an incorrect operation of subtraction.

• Choice 2: $\left(a\times3\right)^{\frac{4}{2}}$ incorrectly divides the exponents.

• Choice 3: $\left(a\times3\right)^{2\times4}$ correctly applies the power of a power rule.

• Choice 4: $\left(a\times3\right)^{2+4}$ adds the exponents instead of multiplying.

The correct answer is clearly Choice 3: $\left(a\times3\right)^{2\times4}$.