Simplify the Nested Expression: ((x×y)^6)^5 Using Laws of Exponents

To solve this problem, we'll follow a structured approach:

• Step 1: Identify the given expression: $\left(\left(x \times y\right)^6\right)^5$.

• Step 2: Apply the Power of a Power rule for exponents.

• Step 3: Simplify the expression to reach the final answer.

Now, let's work through each step:
Step 1: We observe that the expression is $\left(\left(x \times y\right)^6\right)^5$. Here, $(x \times y)$ is raised to the 6th power, and this whole expression is further raised to the 5th power.
Step 2: Apply the Power of a Power rule. This states that if you have an expression $(a^m)^n$, you can simplify it to $a^{m \times n}$.
Therefore, $\left(\left(x \times y\right)^6\right)^5$ becomes $(x \times y)^{6 \times 5}$.
Step 3: Calculate the product of the exponents: $6 \times 5 = 30$. So the expression simplifies to $(x \times y)^{30}$.

Therefore, the solution to the problem is $\left(x \times y\right)^{30}$.

Next, consider the answer choices provided:

• Choice 1: $\left(x \times y\right)^1$ - Incorrect because $6 \times 5 \neq 1$.

• Choice 2: $\left(x \times y\right)^{\frac{5}{6}}$ - Incorrect because $\frac{5}{6}$ does not represent $6 \times 5$.

• Choice 3: $\left(x \times y\right)^{11}$ - Incorrect because $6 \times 5 = 30$, not 11.

• Choice 4: $\left(x \times y\right)^{30}$ - Correct, because the solution matches our simplified expression.