Simplify (x×y)^25 Divided by (x×y)^21: Power Rule Application

We are tasked with simplifying the expression $\frac{\left(x \times y\right)^{25}}{\left(x \times y\right)^{21}}$.

To solve this, we will apply the Power of a Quotient Rule for Exponents. This rule states that when you divide powers with the same base, you subtract the exponents. Formally, $\frac{a^m}{a^n} = a^{m-n}$.

Here, the base $a$ is $(x \times y)$, and the exponents $m$ and $n$ are 25 and 21, respectively. So, applying this rule gives us:

$\frac{\left(x \times y\right)^{25}}{\left(x \times y\right)^{21}} = \left(x \times y\right)^{25 - 21}$

By simplifying the expression $25 - 21$, we get:

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

Thus, the solution to the problem is effectively represented by the expression $\left(x \times y\right)^{25 - 21}$.

Now, looking at the choices provided:

• Choice 1: $\left(x \times y\right)^{\frac{25}{21}}$ is incorrect, as it uses division instead of subtraction.
• Choice 2: $\left(x \times y\right)^{25-21}$ is correct, as it applies the Quotient Rule correctly.
• Choice 3: $\left(x \times y\right)^{25 \times 21}$ is incorrect because it uses multiplication instead of subtraction.
• Choice 4: $\left(x \times y\right)^{25+21}$ is incorrect because it uses addition instead of subtraction.

Therefore, the correct answer is choice 2, $\left(x \times y\right)^{25-21}$.

I am confident in the correctness of this solution based on the consistent application of mathematical rules.