Integer Exponents Rules for Exponents

Unit 1, Lesson 5

Fundamentals

For , three is the exponent or power.Since x is being raised to the exponent, it is the base. • Sometimes parentheses are used to show the

base. • Sometimes exponents are written with ^ such as

x^3 which is the same as

What is the base and what is the exponent for the examples on the next slide:

Examples of Bases and Exponents

a) Base = y; exponent = 4

b) Base = 6; exponent = -1

c) (Base = ; exponent = 8

d) Base = x; exponent = 8

What do you notice is different between examples c & d ?

Rule # 1Multiplying exponents with the same base

(this cannot be used if the bases are different)

Add the exponents:a) = = ∙

b) = = ∙ ∙

c) = ∙ ∙ can’t simplify because the bases are different

d) = = ∙ ∙ ∙ ∙

Rule # 2Dividing exponents with the same base

Subtract the exponents numerator - denominator

a) = = b) = = = (don’t worry about the negative yet; that will be explained)

c) = can’t be simplified b/c the bases are different

Rule # 3Any base raised to zero (0)

Always equal to 1 !The reason is since = 1 (remember anything divided by itself is

1), and also equals = , that means has to equal 1

a) (123)^0 = 1

b) (3xy)^0 =1 (the base is 3xy)

Rule # 4Negative exponents

Take the reciprocal of the base and switch the sign of the exponentEx 1: the base = 5; the reciprocal of 5 is , so = ()^2Ex 2: ()-2 the base is (); the reciprocal is (), so it becomes () 2

Rule # 4 Practice

a) =()4

b) ()-1 = ()1

c) =()2

Rule # 5Raising an exponent to an exponent

Multiply the exponents.

Ex 1: (f3)4 means (f3) times itself 4 times; 4 X3 =12

so, (f3)4 = f 3x4 = f12

Ex 2: ((x2)4)5 = x 2 4 5 ∙ ∙ = x 40

Rule # 6Raising a monomial to an exponent

A monomial is just numbers and variables multiplied together (like 3xy or -5cb). The numbers and variables are connected only through multiplication, no addition or subtraction. In other words, (2x + 5y ) is a binomial two monomials added together; 2x and 5y by themselves are monomialsParentheses help you recognize monomials

Rule # 6Raising a monomial to an exponent

How to do it: raise each number and each variable to the exponent

Ex 1: (3xy)^2 the base is 3xy; the exponent is 2, so: ∙ = 9 ∙

Ex 2: (2 the base is 2; the power is 3: ∙ ()3 = 8∙

Rule # 7Raising a quotient or fraction to an exponent

This is really the same as the previous rule; raise the numerator to the exponent and raise the denominator to the exponentEx 1: ( =

Ex 2: (3 raise 4, d, & to the power of 3

Rule # 8A simple way to deal with negative exponents

This typically is useful when dealing with a fraction that has negative exponents in the numerator, denominator or both.To make the exponent positive, simply move the base from denominator to numerator (or visa versa) and make the exponent positive

Ex 1: =

Rule # 9Raising a fraction to a negative exponent

This is really the same as raising any base to a negative exponent-To make the exponent positive, take the reciprocal of the base

Ex 1: ()-3 = ()3 = =