Calculate (6²)⁷: Solving Nested Power Expression

To solve this problem, we need to simplify the expression $\left(6^2\right)^7$ using the power of a power rule.

The power of a power rule states that when you have an expression of the form $(a^m)^n$, this can be simplified to $a^{m \times n}$.

Let's apply this rule to the given expression:

1. Identify the base and exponents: - Base: $6$ - First exponent (inside parenthesis): $2$ - Second exponent (outside parenthesis): $7$

2. Apply the power of a power rule: - Simplify $(6^2)^7 = 6^{2 \times 7}$.

3. Calculate the final exponent: - Multiply the exponents: $2 \times 7 = 14$. - Therefore, the simplified expression is $6^{14}$.

Considering the answer choices provided:

• Choice 1: $6^{2 \times 7}$ (Correct, as per our solution).
• Choice 2: $6^{2 + 7}$ (Incorrect, addition is used instead of multiplication).
• Choice 3: $6^{7-2}$ (Incorrect, subtraction is used incorrectly).
• Choice 4: $6^{\frac{7}{2}}$ (Incorrect, division is used incorrectly).

Thus, the correct answer to the problem is $6^{2 \times 7}$, which simplifies to $6^{14}$, and aligns with Choice 1.