Evaluate ((7×6)^5)^10: Solving Nested Exponent Expression

To solve this problem, we'll follow these steps:

• Step 1: Analyze the given expression

• Step 2: Apply the appropriate exponent rule

• Step 3: Simplify to reach the final expression

Now, let's work through each step:

Step 1: Begin with the given expression, $\left(\left(7 \times 6\right)^5\right)^{10}$. Here, the inner expression $(7 \times 6)$ is raised to the fifth power, and this result is raised to the tenth power.

Step 2: Use the power of a power rule, which states that $(a^m)^n = a^{m \times n}$. Applying this rule to our expression, we identify $a = (7 \times 6)$, $m = 5$, and $n = 10$.

Step 3: Substitute these values into the formula:

$(7 \times 6)^{5 \times 10} = (7 \times 6)^{50}$

Therefore, the simplified expression is $(7 \times 6)^{50}$.

Upon comparison with the provided answer choices, choice 1 is correct:

• Choice 1: $(7 \times 6)^{50}$ - Correct, matches our simplified result.

• Choice 2: $(7 \times 6)^5$ - Incorrect, doesn't apply exponent rule.

• Choice 3: $(7 \times 6)^2$ - Incorrect, not relevant to problem scope.

• Choice 4: $(7 \times 6)^{15}$ - Incorrect, wrong application of formula.

Therefore, the final answer is $\left(7 \times 6\right)^{50}$.