Exponents and Roots: Break It Down to PRIMES

Say you're eight questions into the Quant section of the Executive Assessment. The screen shows (12^5 × 18^2) ÷ (6^6 × 8). You grab your marker and start multiplying 12 by itself. 144. 1,728. 20,736. 248,832. A minute and a half is gone, and you haven't even touched the 18s yet.

I watch people do this every week, and here's my verdict: you are NOT losing to the math. You're losing to your method. That question is easy. It's just wearing a costume. The costume comes off the second you break every number into PRIMES.

Why primes win

A prime is a whole number bigger than 1 that divides evenly only by 1 and itself: 2, 3, 5, 7, 11, 13 and so on. Every whole number bigger than 1 is built from primes in exactly ONE way. 12 is 2 × 2 × 3. 18 is 2 × 3 × 3. 8 is 2 × 2 × 2. No other recipe exists.

Why does that matter? Because the exponent rules only work when the bases match. The base is the big number. The exponent is the small number that says how many times to multiply it. So 2^3 means 2 × 2 × 2. You can't combine 12^5 with 6^6 directly, because 12 and 6 are different bases. Break both into 2s and 3s, and suddenly EVERYTHING speaks the same language.

Time matters too. The EA Quant section gives you 14 questions, a mix of Problem Solving and Data Sufficiency, in 30 minutes. That's a little over 2 minutes per question on average. Quant is scored 0 to 20 and counts exactly as much as Verbal and Integrated Reasoning. You do not have a minute and a half to spend multiplying 12s.

The only rules you need

• Same base, multiplying: ADD the exponents. 2^3 × 2^4 = 2^7.
• Same base, dividing: SUBTRACT the exponents. 3^9 ÷ 3^6 = 3^3.
• Power of a power: MULTIPLY the exponents. (2^2)^5 = 2^10.
• Different bases, same exponent: combine the bases. 2^3 × 3^3 = 6^3.

That's the whole toolkit. Primes just make sure you always have matching bases to use it on.

Roots want PARTNERS

A square root asks one question: what number times itself gives this? √36 = 6 because 6 × 6 = 36. Break the number into primes and the rule gets simple. Every PAIR of matching primes steps outside the root as a single prime. Anything without a partner stays inside.

Take √72. 72 = 2^3 × 3^2, which is (2 × 2) × (3 × 3) × 2. The pair of 2s comes out as a 2. The pair of 3s comes out as a 3. The lonely 2 stays inside. So √72 = 2 × 3 × √2 = 6√2. Cube roots work the same way, except primes need groups of THREE to escape.

Here's the idea that unlocks a whole family of questions: a whole number is a perfect square (a whole number times itself) exactly when every prime in it shows up an EVEN number of times.

And here's the classic trap. √48 + √75 does NOT equal √123. You cannot add under a root. Break them down instead: 48 = 2^4 × 3, so √48 = 4√3. 75 = 3 × 5^2, so √75 = 5√3. Together that's 9√3.

5 mistakes I see every week

1. Multiplying out big powers. 12^5 by hand is a time sink with a mistake hiding in every step. INSTEAD: factor first. Compute an actual number only at the very end, and only if the answer choices force you to.
2. Stopping at 4, 6 or 9. You write 4^3 × 8^2 and try to combine two different bases. INSTEAD: go all the way down. 4^3 = 2^6 and 8^2 = 2^6, so the product is 2^12. Done.
3. Adding exponents across different bases. 2^3 × 3^2 is NOT 6^5. (It's 72. 6^5 is 7,776.) INSTEAD: add exponents only when the bases match. Combine bases only when the exponents match.
4. Adding under a root. √48 + √75 is not √123. INSTEAD: simplify each root with primes first, then combine only roots with the same thing left inside.
5. Doing it in your head. "I can SEE that 18 is 2 times 9." Sure you can. Until question 12, with four minutes left, when you drop a 3. INSTEAD: write the prime breakdown of every base in your notes, every time, even when it feels beneath you. That feeling is pride, and pride is the most expensive thing you can bring into a test. You draw every geometry figure; you write every prime breakdown.

How to train it

This is training, not reading. A sprinter doesn't think about the start on race day. She drilled it until it was automatic. Your prime breakdowns need to be that automatic.

• Memorize the short list. Powers of 2 through 2^10 (that's 1,024). Powers of 3 through 3^5 (243). Powers of 5 through 5^4 (625). Perfect squares from 1^2 through 15^2 (225).
• Factor everything from 2 to 50. On paper. Until it's boring. Boring means it's automatic.
• Redo old questions. Not new ones. The ones you've already seen. It's not about the right answer or the wrong answer. It's about WHAT YOU'RE DOING to get there. Did you break it into primes, or did you start multiplying?

Stop for a second and think about this. When you got an exponent question wrong in the last month, was it because you didn't know a rule? Or was it because you skipped writing the primes and lost a piece along the way? Don't read on until you have an honest answer.

Here's your challenge for today. Pull up five exponent or root questions you've already done, right or wrong. Start a timer. Redo each one with NOTHING but prime breakdowns on paper. I'll bet at least one of them takes you half the time it took before. Then ask yourself why you were ever doing it the long way.