Three rules instead of a long multiplication
Calculate 2³ · 2⁴. You can do it the hard way: 8 · 16 = 128. But there is a faster path.
2³ is three twos in a multiplication. 2⁴ is four twos. Together there are seven: 2³ · 2⁴ = 2⁷ = 128. You only had to add the exponents.
This is not magic. It is just counting factors. And it always works when the powers have the same base.
Today you will learn three rules: multiplying (you add the exponents), dividing (you subtract the exponents) and a power of a power (you multiply the exponents). Three lines that will save you mountains of multiplying.
One warning right at the start: the rules only work for the same base. 2³ · 5² has no shortcut — you have to calculate both powers separately. Anyone who adds exponents with different bases is making nonsense.
Filip is doing homework: 3⁵ · 3². He is writing out nines and twenty-sixes, the paper fills up. Ema looks at it: “How many threes are you multiplying there?” Filip counts: “Five… and then two more. Seven.” — “So write: 3⁷. Done.” Filip does not believe it, so he checks on small numbers: 3² · 3¹ = 9 · 3 = 27 and 3³ is 27 too. It fits. “So I have been calculating something the whole time that could go on one line?” Ema nods: “That is why you learn the rules.”
When you are not sure about a rule, check it on small numbers: 2² · 2¹ = 4 · 2 = 8 = 2³. Small numbers will always give the rule away.
Three rules for the same base
1. Multiplying: add the exponents. aᵐ · aⁿ = aᵐ⁺ⁿ. Why: m factors and another n factors is m + n factors altogether. Example: 5³ · 5⁴ = 5⁷.
2. Dividing: subtract the exponents. aᵐ : aⁿ = aᵐ⁻ⁿ. Why: division cancels factors. 2⁵ : 2² = (2 · 2 · 2 · 2 · 2) : (2 · 2) — two twos cancel, three are left: 2³. Example: 7⁶ : 7⁴ = 7².
3. A power of a power: multiply the exponents. (aᵐ)ⁿ = aᵐ·ⁿ. Why: (2³)² means 2³ · 2³, which is 3 + 3 = 6 twos. Example: (2³)² = 2⁶ = 64.
The method for every question: first check the base. The same? Use the rule. Different? Calculate the powers separately.
And watch addition: 2³ + 2³ has no exponent rule. That is simply 8 + 8 = 16. The rules work for multiplying and dividing, not for plus and minus.
Example 1: 4² · 4³
The bases are the same (four) → I add the exponents.
4² · 4³ = 4²⁺³ = 4⁵.
If I want the number too: 4⁵ = 4 · 4 · 4 · 4 · 4 = 16 · 16 · 4 = 1024.
Check the other way: 4² = 16, 4³ = 64, 16 · 64 = 1024. It fits. The shortcut and the long way must give the same thing — if they do not, there is a mistake somewhere.
Example 2: 10⁸ : 10⁵
The bases are the same → I subtract the exponents.
10⁸ : 10⁵ = 10⁸⁻⁵ = 10³ = 1000.
It makes sense in plain language too: 100,000,000 : 100,000. You cross out five zeros on top and on the bottom and 1000 is left. Dividing powers of ten is just crossing out zeros — and that is exactly what subtracting the exponents does.
Example 3: (5²)³ vs. 5² · 5³
Two notations that get mixed up:
(5²)³ = 5²·³ = 5⁶ = 15,625. A power of a power → I multiply the exponents.
5² · 5³ = 5²⁺³ = 5⁵ = 3125. Multiplying powers → I add the exponents.
You tell them apart by the brackets. Brackets with an exponent outside = multiply. Two powers next to each other with a dot = add. The results differ by a factor of five — a swap is easy to spot.
The rules only work for the same base. 2³ · 3² is not 6⁵ or anything like it — it is 8 · 9 = 72, calculated separately. Before you reach for a rule, always check the bases.
Practice: now you do it 🎲
Now try it for real. You will see all the questions at once, like on paper. 시작 with ten. When it goes well, add more.
Practise powers — the rules stand on you knowing the basic powers off by heart.
On paper
Write each rule out once as factors. Then you will trust it.
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Write 2³ · 2² out as individual twos and calculate. Then do the same with the shortcut of adding exponents. Compare the results.
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Calculate with the shortcut: 3⁴ · 3³, 8⁶ : 8⁴, (2⁴)². For each one write which rule you used.
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Find the mistake: a classmate wrote 5³ · 5² = 25⁵. Write what they did wrong and how it should be.
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Decide where the shortcut does not work, and calculate by hand: 2³ + 2², 2³ · 3³.
Now you 💪
- I know when I add exponents, when I subtract them and when I multiply them.
- Before I use a rule I check whether the base is the same.
- I can check a rule by writing it out as factors.
Done when: You have all three rules checked by writing them out on paper, the classmate’s mistake corrected, and a run of five correct in practice.
What to take from this lesson
- aᵐ · aⁿ = aᵐ⁺ⁿ — when multiplying you add the exponents.
- aᵐ : aⁿ = aᵐ⁻ⁿ — when dividing you subtract them.
- (aᵐ)ⁿ = aᵐ·ⁿ — a power of a power, you multiply.
- Everything only works for the same base.
- For plus and minus there is no shortcut with exponents.
© 2026 Ing. Martin Polak / AlgoRhino · 콘텐츠 이용 약관