Section 2.8: Exponents & Powers
An exponent indicates how many times a number (the base) is multiplied by itself.
Notation: \( a^n \) means \( a \) multiplied by itself \( n \) times.
- \( 2^3 = 2 \cdot 2 \cdot 2 = 8 \)
- \( 5^0 = 1 \) (any nonzero number to the zero power equals 1)
- \( a^{-n} = \frac{1}{a^n} \)
Rules of Exponents:
- Product: \( a^m \cdot a^n = a^{m+n} \)
- Quotient: \( \frac{a^m}{a^n} = a^{m-n} \)
- Power of a power: \( (a^m)^n = a^{m \cdot n} \)
- Power of a product: \( (ab)^n = a^n b^n \)
Example 1
Simplify \( 2^3 \cdot 2^4 \)
Using product rule: \( 2^3 \cdot 2^4 = 2^{3+4} = 2^7 = 128 \)
Example 2
Simplify \( (3^2)^4 \)
Using power of a power: \( (3^2)^4 = 3^{2 \cdot 4} = 3^8 = 6561 \)
Practice Problems
- Simplify \( 5^3 \cdot 5^2 \)
- Simplify \( \frac{2^7}{2^4} \)
- Simplify \( (4^3)^2 \)
- Simplify \( 3^{-2} \)
- Simplify \( (2 \cdot 5)^3 \)