Simplify (5/xy)⁵: Evaluating Complex Fractions with Fifth Power

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

• Step 1: Recognize the expression as an example of a power of a fraction.
• Step 2: Apply the power of a fraction rule to the expression.
• Step 3: Compare the resulting expression with the answer choices.

Now, let's work through each step:
Step 1: The given expression is $\left(\frac{5}{x \times y}\right)^5$.
Step 2: Apply the power of a fraction rule: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$. This gives us:

$\left(\frac{5}{x \times y}\right)^5 = \frac{5^5}{(x \times y)^5}$

Step 3: Compare with answer choices:

• Choice 1: $\frac{5^5}{(x \times y)^5}$ matches our expression exactly.
• Choice 2: $\frac{5^5}{x^5 \times y^5}$ results from distributing the exponent across the product in the denominator, which is another valid interpretation.
• Choice 3: Indicates both expressions in choices 1 and 2 (a'+b') are correct interpretations of the expanded form.
• Choice 4: $\frac{5^5}{x \times y^5}$ is incorrect as it improperly applies the exponent only to $y$.

Therefore, the solution to the problem, which captures both possible expressions, is a'+b' are correct.