Integrated Math 2 / Unit 1: Exponents and Radicals
1.1 Properties of Exponents
Notes
An exponent tells how many times a base is used as a factor. These rules work when the bases are the same.
- Product rule: $a^m \cdot a^n = a^{m+n}$.
- Quotient rule: $\displaystyle \frac{a^m}{a^n} = a^{m-n}$ (with $a \neq 0$).
- Power rule: $(a^m)^n = a^{mn}$, and $(ab)^n = a^n b^n$.
- Example: $(2x^3)^2 = 2^2 \cdot x^{3 \cdot 2} = 4x^6$.
- Add exponents when multiplying powers. Multiply exponents when raising a power to a power.
Practice
Simplify $x^3 \cdot x^4$.
Same base, so add the exponents: $3 + 4 = 7$.
Simplify $(2x^3)^2$.
Square the 2 to get 4, and multiply the exponents: $3 \cdot 2 = 6$.
Simplify $\displaystyle \frac{12a^7}{3a^2}$.
Divide the coefficients: $12 \div 3 = 4$. Subtract the exponents: $7 - 2 = 5$.
Ready for more? Take the Unit 1 test.