You now have five exponent rules. Each one comes from counting factors, so you can always check a rule with small numbers when you are unsure. This lesson puts them together.
| Rule | Example | Why it works |
|---|---|---|
| $a^{m} \cdot a^{n} = a^{m+n}$ | $x^{2} \cdot x^{5} = x^{7}$ | count all the factors |
| $\dfrac{a^{m}}{a^{n}} = a^{m-n}$ | $\dfrac{x^{6}}{x^{2}} = x^{4}$ | matching factors cancel |
| $(a^{m})^{n} = a^{mn}$ | $(x^{3})^{2} = x^{6}$ | $n$ groups of $m$ factors |
| $(ab)^{n} = a^{n}b^{n}$ | $(2x)^{3} = 8x^{3}$ | every factor is repeated |
| $a^{0} = 1$, $a^{-n} = \dfrac{1}{a^{n}}$ | $x^{-2} = \dfrac{1}{x^{2}}$ | the dividing pattern continues |
Why it works
Every rule is about multiplying, because an exponent is shorthand for repeated multiplication. None of them says anything about adding. $x^{2} + x^{3}$ adds two different kinds of things, like 2 apples plus 3 oranges, so it stays as it is.
Key idea
Simplify one step at a time, from the inside out: powers of powers first, then products, then quotients. Give the final answer with positive exponents.
Worked examples
Worked example: Inside out
Simplify $\dfrac{(x^{3})^{2} \cdot x^{4}}{x^{5}}$. (Assume $x \ne 0$.)
- Power of a power: $(x^{3})^{2} = x^{6}$.
- Multiply on top: $x^{6} \cdot x^{4} = x^{10}$.
- Divide: $x^{10-5} = x^{5}$.
Answer: $x^{5}$
Worked example: Two letters and a negative exponent
Simplify $\dfrac{6a^{-2}b^{5}}{2a^{3}b}$. (Assume no variable is 0.)
- Numbers: $6 \div 2 = 3$.
- $a$: $-2 - 3 = -5$, so $a^{-5} = \frac{1}{a^{5}}$.
- $b$: $5 - 1 = 4$.
- Put it together with positive exponents: $\dfrac{3b^{4}}{a^{5}}$.
Answer: $\dfrac{3b^{4}}{a^{5}}$
Worked example: A value
Find the value of $\dfrac{3^{5} \cdot 3^{-2}}{3^{4}}$.
- Top: $5 + (-2) = 3$, so $3^{3}$.
- Divide: $3 - 4 = -1$, so $3^{-1}$.
- $3^{-1} = \frac{1}{3}$.
Answer: $\frac{1}{3}$
Common mistake
Using a multiplication rule on a sum: $x^{2} + x^{3} = x^{5}$. Check with $x = 2$: $4 + 8 = 12$, but $2^{5} = 32$. Only like terms combine when you add, and then the exponent stays put: $3x^{2} + 4x^{2} = 7x^{2}$.