Test 2 Section 3 #7

This is one of my favorite questions because there are a couple fun paths to take: one with pure exponent rules, and one with exponent rules and difference of two squares. So fun! (I understand if you don’t share my enthusiasm.)

First, the exponent-rules-only method. We need to take what we’re given and simplify it enough that we can stop worrying about *x* and focus exclusively on *a* and *b*. To do that, we first need to remember that when we raise a power to a power, we multiply the powers. In other words:

From there, we need to remember that when we divide exponential expressions with the same base, we *subtract* the exponents.

From there, we’re good to eliminate *x* and just say that . Since the question asks us for the value of , just divide that equation by 2 to get .

Going the second way involves the division exponent rule but not the power-to-a-power rule. Skip right to the division rule:

From there, we can say that and use our difference of two squares skills.

The question tells us that , so let’s substitute:

So there you go—two fun ways to get this one done. Which do you prefer?