Exponent Rules
Five rules that every student copies onto a card and half of them then apply wrongly. Written out as repeated multiplication, none of them needs remembering at all.
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Everybody copies the exponent rules onto a card, and a term later half of them are being applied to the wrong thing. That is not a memory failure. A rule you have only memorised gives you no way to tell whether it applies, so it gets used everywhere it superficially fits.
The fix is to stop treating them as rules. Pick one below and read the middle line.
The middle line is the proof. Every one of these rules is obvious once the powers are written out as repeated multiplication - which means you never have to trust your memory for them.
Every rule is a counting argument
a^3 is shorthand. What it actually says is a · a · a - three copies of a, multiplied together. Once both sides of a rule are written out that way, the rule stops being a claim and becomes a count.
Multiplying: two copies next to three copies is five copies.
a^2 · a^3 = (a·a)(a·a·a) = a^5
You did not add the exponents because a rule said so. You added them because you were counting how many a's ended up in the line.
Dividing: five copies over two copies, and two of them cancel.
a5/a2 = (a·a·a·a·a)/(a·a) = a^3
A power of a power: three copies of "two copies" is six copies.
(a2)3 = (a·a)(a·a)(a·a) = a^6
Three rules, one argument. The reason people mix up "add" and "multiply" is that they memorised the verbs instead of the count.
Zero and negative exponents are forced, not chosen
These two feel like conventions somebody picked, and they are not.
Take a3/a3. Anything over itself is 1. But the quotient rule says it is a^(3-3) = a^0. Both describe the same division, so a^0 = 1 is the only value that keeps the rules consistent.
Keep going down. Each step down the exponent divides by a once more:
| power | value when a = 2 |
|---|---|
| 2³ | 8 |
| 2² | 4 |
| 2¹ | 2 |
| 2⁰ | 1 |
| 2⁻¹ | 1/2 |
| 2⁻² | 1/4 |
Nothing changes direction at zero - the halving just carries on. That is all a negative exponent is: a^-n = 1/a^n. In particular it is not a negative number. 2^-3 is 1/8, which is positive.
The three mistakes worth naming
Different bases do not combine. 2^3 · 3^4 stays as it is. The rules are about counting copies of the same thing, and these count two different things.
Adding terms is not multiplying them. x^2 + x^3 is not x^5. Write it out: two copies plus three copies is not five copies, it is a sum of two different quantities. Nothing cancels, nothing combines.
Powers distribute over products, not over sums. (ab)^2 = a2b2 is true - two copies of ab is two a's and two b's. But (a+b)^2 is not a^2 + b^2; it expands to a^2 + 2ab + b^2, and the middle term is exactly what the distributive property predicts.
Try one
If you reached for a calculator, try it again by cancelling instead: four copies of 3 over two copies of 3 leaves two copies, and two copies of 3 is 9. That is the version that keeps working when the base is x rather than 3.
Where this goes next
Exponent rules are what make scientific notation arithmetic work, what turn a root into a fractional power, and what let a logarithm turn multiplication into addition. All three are the same five rules, read in a different direction - which is a good reason to have them by argument rather than by heart.
Common questions
- What are the exponent rules?
- Five that carry almost everything: multiplying powers of the same base adds the exponents, dividing subtracts them, raising a power to a power multiplies them, anything to the power 0 is 1, and a negative exponent means one over the positive one. All five are consequences of writing the powers out as repeated multiplication.
- Why is anything to the power of zero equal to 1?
- Because of the quotient rule. Any number divided by itself is 1, and the quotient rule says the same division is the base to the power of the exponent minus itself, which is 0. Both statements describe the same division, so a to the power 0 has to be 1. It is forced, not decreed - the one exception being 0 to the power 0, which is left undefined.
- What does a negative exponent mean?
- It means a reciprocal, not a negative number. 2 to the power -3 is 1/8, which is positive. Keep dividing by the base as the exponent falls - 8, 4, 2, 1 - and continuing past 1 gives 1/2, 1/4, 1/8. A negative exponent tells you how many times to divide instead of multiply.
- When can I add exponents and when can I not?
- Only when the bases are identical and you are multiplying. 2 cubed times 2 to the fourth is 2 to the seventh, but 2 cubed times 3 to the fourth cannot be combined at all - they count different things. Adding exponents while adding terms is the single most common mistake here: x squared plus x cubed is not x to the fifth.
- Is (a + b) squared equal to a squared plus b squared?
- No, and this is the mistake worth naming. Squaring distributes over multiplication, not over addition. (ab) squared really is a squared times b squared, but (a + b) squared expands to a squared plus 2ab plus b squared. The missing middle term is what the FOIL box makes visible.
Now do it yourself, properly
Reading someone else's working is not the same as being able to do it. The Coddy math course puts you on a board that checks every move you make, so you find out where you actually stand.
Want to run this on your own numbers?
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