Solve ((7×2)³)⁻² : Nested Exponents with Negative Powers

To solve the expression $\left(\left(7 \times 2\right)^3\right)^{-2}$, follow these steps:

• Step 1: Simplify the expression inside the first parentheses. $7 \times 2 = 14$
• Step 2: Apply the power of a power rule. $\left(14^3\right)^{-2}$ becomes $14^{3 \times (-2)}$
• Step 3: Calculate the exponent. $3 \times (-2) = -6$
• This gives us: $14^{-6}$
• The expression $\left(14^3\right)^{-2}$ simplifies to $\left(7 \times 2 \right)^{-6}$.

The expression simplifies to $\left(7 \times 2\right)^{-6}$.

Now, let's match this with the given choices:

• Choice 1: $\left(7 \times 2\right)^{-6}$ - This matches our result and is the correct answer.
• Choice 2: $\left(7 \times 2\right)^{\frac{-2}{3}}$ - Incorrect, since the power of a power rule wasn’t applied correctly.
• Choice 3: $\left(7 \times 2\right)^{-1}$ - Incorrect simplification of the given problem.
• Choice 4: $\left(7 \times 2\right)^1$ - Incorrect simplification showing misunderstanding of negative exponent rules.

Therefore, the correct answer is Choice 1: $\left(7 \times 2\right)^{-6}$.