1. Definition of Let b represent any real number and n represent a positive integer. Then, n factors of b

2. Multiplying with Like Bases: The Product Rule For any number a and any positive integers m and n, (When multiplying, if the bases are the same, keep the base and add the exponents.)

3. Multiplication of Like Bases Assume that a≠ 0 is a real number and that m and n represent positive integers. Then, Property 1

4. Multiply and simplify: Multiplying powers by adding exponents Using the associative and commutative laws Multiplying coefficients; using the product rule

5. Simplifying More Expressions (3p²q⁴)(2pq⁵) Apply the associative and commutative properties of multiplication to group coefficients and like bases =(3∙2)(p²p)(q⁴q⁵) Add the exponents when multiplying like bases. Simplify

6. Division of Like Bases Assume that a ≠ 0 is a real number and that m and n represent positive integers such that m > n. Then, Property 2:

7. Simplifying Expressions with Exponents (t∙t∙t∙t∙t∙t) (t∙t∙t∙t) Subtract the exponents 5∙5∙5∙5∙5∙5 5∙5∙5∙5 Subtract the exponents ( the base is unchanged).

8. Simplifying Expressions with Exponents Subtract the exponents Note that 10 is equivalent to 10¹ Add the exponents in the denominator ( the base is unchanged). Add the exponents in the numerator ( the base is unchanged). Subtract the exponents Simplify

9. Simplifying More Expressions Group like coefficients and factors. Subtract the exponents when dividing like bases. Simplify

10. Definition of Let b be a nonzero real number. Then, Definition of Let n be an integer and b be a nonzero real number. Then,

11. Patterns 3³ = 27 As the exponents decrease by 1, the resulting expressions are divided by 3 3² = 9 3¹ = 3 3° = 1 For the pattern to continue, we define 3° = 1

12. Simplifying Expressions with a Zero Exponent Simplify 4° = 1 By Definition (-4)° = 1 By Definition -4° = -1The exponent 0 applies only to 4 z° = 1 By Definition -4z° = -4∙z° = -4∙1 = -4 The exponent 0 applies only to z. (4z)° = 1 The parentheses indicate that the exponent, 0, applies to both factors 4 and z

13. Definition of 3³ = 27 3² = 9 3¹ = 3 3° = 1 As the exponents decrease by 1, the resulting expressions are divided by 3 When the exponent is negative take the reciprocal of the base and change the sign of the exponent

14. By Definition Simplify The base if -3 and must be enclosed in parentheses. (-3)(-3)(-3)(-3) = 81 Take the reciprocal of the base, and change the sign of the exponent. Simplify Take the reciprocal of the base, and change the sign of the exponent. Simplify Apply the power of a quotient rule. Take the reciprocal of the base, and change the sign of the exponent. Apply the exponent of 3 to each factor within parentheses. Simplify Note that the exponent, -3, applies only to x.

15. Class fun

16. Power Rule for Exponents Assume the a ≠ 0 is a real number and that m and n represent positive integers. Then, Property 3

17. Simplifying Expressions with Exponents Multiply exponents (the base is unchanged) Simplify inside the parentheses by adding exponents Multiply exponents (the base is unchanged)

18. Power of a Product and Power of a Quotient Assume that a and b are real numbers such that b ≠ 0. Let m represent a positive integer. Then, Property 4: Property 5:

19. Raise each factor within parentheses to the fourth power Raise each factor within parentheses to the third power. Multiply exponents and simplify Square each factor within parentheses Multiply exponents and simplify

20. Clear parentheses by applying the power rule Multiply exponents. Add exponents in the numerator. Subtract exponents Try these expressions