Powers: Reverse Rules - Dividing Powers with the same Base

Which of the following expressions is equal to $$50 \cdot 5^{a-2}$$?

Incorrect. Did you reach this answer choice by multiplying 50 by 5?

Correct. [[snippet]] Apply the reverse rule for dividing powers to transform $$5^{a-2}$$ into $$ \frac{5^a}{5^2}$$. >$$ 50 \cdot 5^{a-2}= 50 \cdot \frac{5^a}{5^2} = 50 \cdot \frac{5^a}{25}$$ Next, reduce 25 with the 50 in the numerator, leaving $$ \frac{2}{1}$$. >$$ 50 \cdot \frac{5^a}{25}= 2 \cdot \frac{5^a}{1} = 2 \cdot 5^a$$

Incorrect. [[snippet]]

A common, and dangerous, mistake. Note that you can't multiply a power by another term by multiplying the base alone. In this case, $$50~ \cdot$$ [power of 5] is not the same as $$50 \cdot 5$$. In fact, $$5^a$$ is actually $$5 \cdot 5 \cdot 5…$$ a total of $$a$$ times. You cannot just multiply 50 times a single 5 and ignore the power. Multiplying a power by a coefficient demands resolving the power first—or at least breaking it apart into something more manageable. The golden rule of powers is very simple: if it isn't defined as one of the power rules (i.e., multiplication = add the exponents, division = subtract the exponent, and so on), don't do it. [[snippet]]

[[snippet]]

$$5^a$$

$$2 \cdot 5^a$$

$$250^a$$

$$250^{a-2}$$

Yes.

No.