Solve ((8×9)^11)^4: Multiple Exponents with Nested Parentheses

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

• Step 1: Identify the base and the exponents in the given expression.
• Step 2: Use the power of a power rule to simplify the expression.
• Step 3: Verify the solution against given answer choices.

Now, let's work through each step:

Step 1: The given expression is $((8 \times 9)^{11})^4$. Here, the base is $8 \times 9$, and the original exponent of the entire base is $11$. There is an outer exponent of $4$.

Step 2: Apply the power of a power rule, $(a^m)^n = a^{m \cdot n}$.
Thus, $((8 \times 9)^{11})^4 = (8 \times 9)^{11 \cdot 4}$.

Step 3: Perform the multiplication of exponents:
$11 \cdot 4 = 44$.
Therefore, $((8 \times 9)^{11})^4 = (8 \times 9)^{44}$.

Therefore, the solution to the problem is $(8 \times 9)^{44}$.

Now let's check the provided answer choices:

• Choice 1: $(8 \times 9)^{15}$ - Incorrect, as the operation is $(11 \times 4)$, not $(11 + 4)$.
• Choice 2: $(8 \times 9)^{44}$ - Correct, since $11 \times 4 = 44$.
• Choice 3: $(8 \times 9)^{\frac{4}{11}}$ - Incorrect, as this result is unrelated to multiplied exponents.
• Choice 4: $(8 \times 9)^7$ - Incorrect, as there is no reason to have a resulting exponent of $7$.

Therefore, the correct choice is Choice 2: $(8 \times 9)^{44}$.