Section 5.1

Product and Power Rules for Exponents

5.1 Lecture Guide: Product and Power Rules for Exponents

Objective 1: Convert between exponential form and expanded form.

Algebraically

For any natural number n,

with base b and exponent n.

Verbally

For any natural number n, is the product of b used as a ____________ n times. The expression is read as “b to the nth power.”

Numerical Example

Exponential Notation

nb

factors of

n

n b

b b b b

35 5 5 5

24 4 4

Write each expression in exponential form.

1. 2.

3. 4.

x x x y y 2 x x x x

x x x y y y x y x y x y

Write each exponential expression in expanded form.

5. 6.3( 5 )a35a

Write each exponential expression in expanded form.

7. 8.3 2a b 27

Write each exponential expression in expanded form.

9. 10. 2a b3 2a b

11. Complete the warm-up examples below:

Expanded Form:

2 5

__________________

______

x x x x x x x x x

Alternate Form:

2 5 2 5

______

x x x

Algebraically

For any real number x and natural numbers m and n,

Verbally

To multiply two factors with the same base, use the common base and ____________ the exponents

Algebraic Example

_________________

Product Rule for Exponents

.m n m nx x x

2 4x x

12. 13.

Simplify Each Expression

4 3m m 10 12b b

14. 15.

Simplify Each Expression

4 6 23r r r 3 52 3x x

16.

Expanded Form:

Alternate form:

Objective 3: Use the power rule for exponents.

Complete the warm-up examples below:

32 2 2 2

__________________

______

x x x x

32 2 3

______

x x

Algebraically

For any real number x and natural numbers m and n,

Verbally

To raise a power to a power, ____________ the exponents

Algebraic Example

_________________

Power Rule for Exponents

( ) .m n mnx x

2 5( )x

17. 18.

Simplify Each Expression

43x 32x

19. 20.

Simplify Each Expression

23x 23x

21.

Expanded Form:

Shortcut:

Complete the warm-up examples below:

3

_________

xy xy xy xy

xxx yyy

3_________xy

Algebraically

For any real numbers x and y and any natural number, m,

Verbally

To raise a product to a power, raise each ___________ to this power.

To raise a quotient to a power, raise both the ________ and the _________ to this power.

Algebraic Example

_________________

_________________

Raising Products and Quotients to a Power

,m m mxy x y

for 0.m m

m

x xy

y y

5xy

5x

y

22. 23.

Simplify Each Expression

3mn 2 4( )x y

24. 25.

Simplify Each Expression52

3

mn

3xy

26. 27.

Simplify Each Expression

42x 4

2x

28. 29.

Simplify Each Expression

2

3

7x

23

4

35ab

30. Evaluate the expression for 2 4b ac 4,a 2,b 3.c and

Comparing Addition and Multiplication

Add the like terms and simplify the products.

31. 32.2 2 2x x x 2 2 2x x x

33. 34.4 43 5x x 4 43 5x x

Comparing Addition and Multiplication

Add the like terms and simplify the products.

35. 36.If possible, add If possible, multiply 2 32 5x x 2 32 5x x

Comparing Addition and Multiplication

Add the like terms and simplify the products.

37. The formula for the area of a square is where x is the length of a side of the square.

(a) Complete the table of values for the area of a square with sides of length x cm.

Length of a side, x cm Area, cm2

1

2

3

4

2A x

37. The formula for the area of a square is

(b) How does the area of a square with sides of length 2 cm compare to the area of a square with sides of length 4 cm?

2 cm

4 cm

2A xwhere x is the length of a side of the square.