Complete the Expression: (x×y)^4a Power Rule Problem

Insert the corresponding expression:

$\left(x\times y\right)^{4a}=$

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

• Step 1: Identify the given expression as $(x \times y)^{4a}$.
• Step 2: Apply the power of a power rule $(a^m)^n = a^{m \times n}$.
• Step 3: Transform the expression using these mathematical rules.

Now, let's work through each step:

Step 1: The problem provides us with the expression $(x \times y)^{4a}$. This expression involves a product raised to a power $4a$.

Step 2: We aim to express this using the idea of a power raised to another power. According to this rule, we can interpret $(x \times y)^{4a}$ as $((x \times y)^4)^a$. The rule applied here is $(a \times b)^n = a^n \times b^n$ in reverse, leading to $(a^m)^n = a^{m \times n}$ understanding for breaking down.

Step 3: Thus, $(x \times y)^{4a}$ becomes $((x \times y)^4)^a$.

After analyzing the answer choices:

• Choice 1, $\left(x \times y\right)^4 \times \left(x \times y\right)^a$, does not use the power of a power rule, it is a product.
• Choice 2, $\left((x \times y)^4\right)^a$, correctly represents the power of a power rule.
• Choice 3, $\frac{\left(x \times y\right)^4}{\left(x \times y\right)^a}$, represents division of powers, not the intended structure.
• Choice 4 cannot be correct as it lists combinations that do not follow from the given rules.

Therefore, the correct solution to the problem is $\left((x \times y)^4\right)^a$, which is answer choice 2.