Solve Nested Powers: Calculating (3²)⁴ Using Exponent Rules

To solve this problem, we'll utilize the Power of a Power rule of exponents, which states:

$(a^m)^n = a^{m \cdot n}$

Given the expression $(3^2)^4$, we need to simplify this by applying the rule:

• Step 1: Recognize that we have a base of 3, with an exponent of 2, raised to another exponent of 4.
• Step 2: According to the Power of a Power rule, we multiply the exponents: $2 \times 4$.
• Step 3: Compute the product of the exponents: $2 \times 4 = 8$.
• Step 4: Rewrite the expression as a single power: $3^8$.

This simplifies the original expression $(3^2)^4$ to $3^{8}$.

Comparing this with the given choices:

• Choice 1: $3^{2 \times 4}$ is equivalent to $3^8$, confirming it matches our solution.
• Choices 2, 3, and 4 involve incorrect operations with exponents (addition, subtraction, division) and therefore do not align with the necessary Power of a Power rule.

Thus, the correct answer to the problem is:

$3^{8}$, and this corresponds to Choice 1: $3^{2 \times 4}$.