# A Common Sense Approach to Exponents

Memory is a fickle thing. Most of us have no problem remembering the lyrics to some song that we were really into in 8th grade, but from time to time forget where we put our car keys. Even worse, our memories work the worst when we need them the most. So in a stressful situation that can go a long way to determining your future, why would you want to have to rely on your memory?

Some memorizing is necessary, but as I help my students prepare for the GMAT (or GRE) my approach is always to explain the reasoning behind a rule, or the logic behind a particular formula. If you can’t remember the formula for the area of a trapezoid, you’ll still have a chance to figure out the correct answer by remembering that a trapezoid can be broken down to a rectangle and triangles.

Coming back to the internal logic of a rule rather than relying solely on memory is something that’s particularly helpful with exponent rules. What would you do when faced with the following problem?

$\dpi{200}[(a^2)(a^3)]^4=?$

Working first inside the largest set of parenthesis we have multiplied by . Now for many of you, that doubt is creeping into your mind: Do I multiply or add those exponents? You may have memorized exponent rules at some point, but how can you be sure that you’re remembering them correctly?

Rather than obsessing about the accuracy of you memory, a quick check of this problem can be done by thinking about what exponents mean. When we say what we mean is a multiplied by itself three times. If we take that basic understanding and apply it to this problem we get:

$\dpi{200}[(a \times a) \times (a \times a \times a)]^4=?$

Now it’s clear that we have a multiplied by itself 5 times, which is a5. That fits perfectly with the relevant exponent rule, with no cause for doubt. To resolve the rest of this problem we can use the same approach. Raising something to the fourth power means multiplying it by itself four times, so we get a5 times itself five times, then another five, then another five, then another five for a total of 20.

$\dpi{200}[(a \times a) \times (a \times a \times a)]^4=a^5 \times a^5 \times a^5 \times a^5 =a^{20}$

When you feel overwhelmed by the number of things you need to memorize to prepare for your GMAT, remember that taking a little extra time to understand the logic behind those rules you’re memorizing puts a lot less reliance on your fickle memory and puts you in a much better position to score well.