# Basic Algebra/Polynomials/Exponents

## Vocabulary[edit]

**Base**: The number directly preceding an exponent

EX: a^{2} -> a is the base

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

EX: a^{4} = a * a * a * a -> 4 is the exponent

EX: 4^{3} = 4 * 4 * 4 = 64 -> 3 is the exponent

## Lesson[edit]

Exponents are a simple way to represent repeated multiplication. For example a x a = a^{2}. 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 = a^{1}

2) When we multiply the same base we add our exponenents: a^{3} x a^{2} = a^{3 + 2} = a^{5}

3) When we divide the same base we subtract our exponents: a^{6} / a^{4} = a^{6 - 4} a^{2}

4) When we raise a power to a power we multiply our exponents: (a^{2})^{3} = a^{2 * 3} = a^{6}

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

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

## Example Problems[edit]

x^{3} * x^{6} = ?? [= x^{9}]

a^{x} * a^{3} = ?? [= a^{x + 3}]

x^{4} / x^{2} = ?? [= x^{2}]

(a^{4}b)^{3} = ?? [= a^{12}b^{3}]

(a^{2}b^{3}c)^{3} = ?? [= a^{6}b^{9}c^{3}]