When you multiply powers with the same base, do you add, subtract, or multiply the exponents? Apply that rule to the $a$ terms, then to the $b$ terms, then to the $c$ terms.

When you add a positive and a negative exponent, think of it like adding positive and negative integers. Pay special attention to the $c$ terms: one has exponent $1$ and the other has exponent $-2$.

In Desmos, pick simple positive values for the variables, for example type a=2, b=3, and c=5. (Any positive values work, but small integers are easiest.)

Type the original expression (a^2*b^-3*c)*(a^-1*b^5*c^-2) and note the numerical result that Desmos gives.

One by one, type each answer choice using the same a, b, and c values (for example a*b^2*c^-1, a^3*b^2*c, etc.). The choice whose value matches the value of the original expression is the equivalent expression.

Start by rewriting the product so that the same bases are next to each other:

Now you can combine the exponents for each base separately.

Recall: when you multiply with the same base, you add the exponents.

• For $a$:
  $a^{2} \cdot a^{-1} = a^{2 + (-1)} = a^{1}$.
• For $b$:
  $b^{-3} \cdot b^{5} = b^{-3 + 5} = b^{2}$.
• For $c$:
  $c^{1} \cdot c^{-2} = c^{1 + (-2)} = c^{-1}$.

So the expression becomes a product of these three simplified factors.

Putting the simplified pieces together:

Compare this with the answer choices to see it matches choice A) $ab^{2}c^{-1}$.