Simplify (8×6)^(-7))^(-8): Nested Negative Exponents Problem

To solve this problem, we'll apply the power of a power rule for exponents. The problem is to simplify the expression $\left(\left(8\times6\right)^{-7}\right)^{-8}$.

• Step 1: Understand the Power of a Power Rule

The power of a power rule states that $(a^m)^n = a^{m \times n}$.

• Step 2: Apply the Rule

Here, the base of the entire expression is $8 \times 6$, the first exponent is $-7$, and the second exponent is $-8$. According to the rule, we multiply the exponents:

$(8 \times 6)^{-7 \times -8}$

• Step 3: Simplify the Exponent Calculation

Calculate the multiplication of the exponents:

$-7 \times -8 = 56$

This results in the expression:

$(8 \times 6)^{56}$

Considering the given choices, carefully cross-check against our simplified expression:

• Choice 1: $\left(8\times6\right)^{-7-8}$ is incorrect - it uses addition instead of multiplication.

• Choice 2: $\left(8\times6\right)^{\frac{-8}{-7}}$ is incorrect - it uses division instead.

• Choice 3: $\left(8\times6\right)^{-7+8}$ is incorrect - it uses addition as well.

• Choice 4: $\left(8\times6\right)^{-7\times-8}$ is our correct transformation before final simplification.

After calculating, the expression $\left(8\times6\right)^{-7\times-8} = (8 \times 6)^{56}$. The corresponding expression reflects Choice 4 before final simplification.