Simplify (16×5)^25 ÷ (16×5)^21: Exponent Division Problem

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

• Step 1: Identify the expression's base and exponents.
• Step 2: Apply the quotient rule for exponents.
• Step 3: Simplify the expression.

Now, let's work through each step:

Step 1: The given expression is $\frac{(16 \times 5)^{25}}{(16 \times 5)^{21}}$, where the base is $(16 \times 5)$ and the exponents are 25 and 21.

Step 2: Applying the quotient rule for exponents, which states that $\frac{a^m}{a^n} = a^{m-n}$, we subtract the exponent of the denominator from the exponent of the numerator. Thus, we have:

$(16 \times 5)^{25-21}$

Step 3: Simplifying the exponents results in:

$(16 \times 5)^4$

Therefore, the correct choice is the simplified expression:

$(16 \times 5)^{25-21}$

Upon reviewing the provided choices:

• Choice 1: $(16 \times 5)^{25 \times 21}$ is incorrect because it indicates multiplication of exponents, not the subtraction needed.
• Choice 2: $(16 \times 5)^{25-21}$ is correct, reflecting the exponent subtraction.
• Choice 3: $(16 \times 5)^{25+21}$ is incorrect as it applies the wrong rule (addition instead of subtraction).
• Choice 4: $(16 \times 5)^{\frac{25}{21}}$ is incorrect as the quotient rule for exponents requires subtraction, not division.

Thus, choice 2 is the correct answer.