5.1 Exponents

Exponents that are natural numbers are shorthand notation for repeating factors. 3 4 = is the base 4 is the exponent (also called power) Note by the order of operations that exponents are calculated before other operations.

Evaluate each expression. a. 3 4 b. (–5) 2 c. –6 2 d. (2 4) 3 e Example

Evaluate each expressions for the given value of x. Example a. Find 3x 2 when x = 5. b. Find –2x 2 when x = –1.

The Product Rule for Exponents If m and n are positive integers and a is a real number, then a m · a n = a m+n

Use the product rule to simplify. a. 3 2 · 3 4 b. x 4 · x 5 c. z 3 · z 2 · z 5 d. (3y 2 )(–4y 4 ) Example

The Power Rule If m and n are positive integers and a is a real number, then (a m ) n = a mn

Use the power rule to simplify. a. (2 3 ) 3 b. (x 4 ) 2 Example

If n is a positive integer and a and b are real numbers, then (ab) n = a n · b n Power of a Product Rule Example: (5x 2 y) 3

Examples Simplify each expression. a. b.

If n is a positive integer and a and c are real numbers, then Power of a Quotient Rule Example:

Example Simplify the expression.

Quotient Rule for Exponents Example: If m and n are positive integers and a is a real number, then

Example Simplify the expression. Group common bases together.

a 0 = 1, as long as a is not 0. Note: 0 0 is undefined. Example: a. 5 0 = b. (xyz 3 ) 0 c. –x 0 Zero Exponent