Vocabulary

Base: The number directly preceeding an exponent

EX: a² -> a is the base

Exponent: The number (written in superscript) used to express how many times a base is multiplied by itself

EX: a⁴ = a * a * a * a -> 4 is the exponent

EX: 4³ = 4 * 4 * 4 = 64 -> 3 is the exponent

Lesson

Exponents are a simple way to represent repeated multiplication. For example a x a = a². There are a few simple rules for exponents that help reduce very large problems to simple little ones. The rules are as follows:

1) The exponent of any number is always a one (1): a = a¹

2) When we multiply the same base we add our exponenents: a³ x a² = a^{3 + 2} = a⁵

3) When we divide the same base we subtract our exponents: a⁶ / a⁴ = a^{6 - 4} a²

4) When we raise a power to a power we multiply our exponents: (a²)³ = a^{2 * 3} = a⁶

5) When we raise a PRODUCT to a power we raise both parts of the product to the power: (ab)³ = a³b³ [NOTE: This ONLY works with multiplication and NOT addition: (a + b)³ ≠ a³ + b³]

6) When we raise a QUOTIENT to a power we raise both parts of the quotient to the power: (a/b)² = a² / b² [NOTE: This ONLY works with division and NOT subtraction: (a - b)² ≠ a² - b²

Example Problems

x³ * x⁶ = ?? [= x⁹]

a^x * a³ = ?? [= a^{x + 3}]

x⁴ / x² = ?? [= x²]

(a⁴b)³ = ?? [= a¹²b³]

(a²b³c)³ = ?? [= a⁶b⁹c³]