Exponents: Break EVERYTHING Down to Primes

Here is a scene I watch play out every week. A GRE Test Taker sees 4^x × 8^3 = 2^15, panics a little, and clicks open the on-screen calculator. Thirty seconds later they are punching in 8 × 8 × 8, staring at 512, and wondering what to do with the 4^x. A minute later they guess.

That question was never hard. It just LOOKED hard, because the bases did not match. (A base is the big number. The exponent is the little number up top that tells you how many times to multiply the base by itself. Since we can't print little raised numbers here, 2^15 means 2 to the 15th power.)

My verdict is blunt: if you are not breaking EVERY number in an exponent question down to primes, you are making these questions harder than they are. A prime is a whole number greater than 1 that can only be divided evenly by 1 and itself: 2, 3, 5, 7, 11 and so on. Every whole number bigger than 1 is built out of primes. 4 is 2 × 2. 8 is 2 × 2 × 2. 12 is 2 × 2 × 3. Once everything is written in primes, the ugly question turns into counting.

The only three rules you need

Before the examples, get these on paper. Not in your head. On paper.

• Same base, multiplying: ADD the exponents. 2^3 × 2^4 = 2^7.
• Same base, dividing: SUBTRACT the exponents. 3^6 ÷ 3^2 = 3^4.
• Power of a power: MULTIPLY the exponents. (2^2)^5 = 2^10.

Notice what all three have in common: the words "same base." That is the whole game. The rules only work when the bases match, and primes are how you MAKE them match.

Example 1: the one from the opening

Stop for a second. How long did that take you once 4 and 8 became 2s? I'll bet it was under a minute. The question didn't change. Your approach did.

Example 2: a Quantitative Comparison

On the GRE, a Quantitative Comparison question gives you two quantities and asks how they relate. The four answer choices are always the same, so learn them once and you never have to read them again.

Example 3: when there's a variable, TEST A VALUE

Which way was faster for you? Doesn't matter. What matters is that you had TWO ways, and both of them were on paper.

5 mistakes I see all the time

1. Reaching for the calculator first. The GRE does give you an on-screen calculator, and it's useful for ugly division. It's terrible for 8^7. You will mistype, and the numbers get too big to trust. INSTEAD: break the bases into primes before you even think about the calculator. Most of the time you won't need it.
2. Combining exponents when the bases don't match. 2^3 × 3^2 is NOT 6^5. It's 8 × 9 = 72. The adding rule only works on the same base. INSTEAD: ask "are these bases the same?" every single time. If not, go to primes.
3. Breaking down SOME of the numbers. Converting the 4 but leaving the 8 alone means you still have mismatched bases. INSTEAD: break down EVERYTHING. Every base, top and bottom, both quantities. No exceptions.
4. Mixing up adding and multiplying exponents. 2^2 × 2^5 is 2^7. (2^2)^5 is 2^10. Mix those up and you land right on a trap answer. INSTEAD: write the rule next to your work the first few weeks until it's automatic.
5. Doing it in your head. "I can just see that 9 is 3 squared." Sure. And then you forget to double the exponent. That's pride, and pride is expensive. INSTEAD: write every conversion on your scratch paper. It takes five seconds and saves you the silly mistake.

If exponents keep tripping you up on inequalities too, watch out for negative bases. (−2)^2 is 4, but (−2)^3 is −8, and the GRE loves to hide that sign. The same habit of breaking things into small pieces also pays off when you count outcomes in probability.

Why this matters more than it looks

Exponent questions show up across the Quant section, which, per ETS's description of Quantitative Reasoning, includes arithmetic topics like exponents and roots alongside algebra and geometry. You get 35 minutes for each 20 question Quant section. Every exponent question you finish in under a minute is time you get to spend on the wordy questions that actually need it.

And here's the bigger point: the GRE is not testing whether you can multiply 8 by itself seven times. It's testing whether you can see a structure under the mess. Primes are that structure. Train the habit and the "hard" questions stop being hard.

Your challenge for today

Pull 10 exponent questions from whatever practice material you're using, including ones you've already done. Redo every one of them with a timer running and a rule: EVERY base gets broken into primes on paper before you do anything else. No calculator until the primes are written. Then compare your times and your accuracy to the first time you did them. I'll bet both got better. Do it tonight.