6.1 Properties of Exponents

1/8/2014

Power, Base and Exponent:

73

Exponent: is the number that tells you how many times the base is multiplied to itself.

In this example 73 means 7•7•7

Product of Powers:

Ex. 32 • 35

= 3•3•3•3•3•3•3 = 37

= 32+5

In general: am•an = am+n

Note: bases must be the same.

Power of a Power:

Ex. (23 )2

= (23 )• (23 )=(2•2•2)•(2•2•2)= 26 = 23•2

In general: (am)n = am•n

Power of a Product:

Ex. (4•x )3

= (4•x )• (4•x ) • (4•x ) =(4•4•4)•(x•x•x) = 4 3 •x 3

In general: (a •b)m =a m•b m

Note: Distribute the outside exponent to the inside exponent.

Zero Exponent

In general: a0 = 1

44 164 644 123

55 255 1255 123

33 93 273 123

÷5 ÷5 ÷5

÷4 ÷4 ÷4

÷3÷3 ÷3

Any base raised to a 0 power equals 1.

50 = 1

40 = 1

30 = 1

3

Exponential Form Fraction Form

33 27

32 9

31 3

30 1

3-1

3-2 =

3-3 =

What happens when a base is raised to a negative power?The power moves from top to bottom and the exponent becomes positive!

Negative Exponent

Ex. 5-2

Ex.

In general:

251

mm

aa 1

Negative exponent MOVES power. If the power with a negative exponent is in the numerator, the power moves to the denominator and exponent becomes positive. If the power with a negative exponent is in the denominator, the power moves to the numerator and exponent becomes positive.

33 22 x

x

Quotient of Powers

Ex.

In general:33

3333333

2

5

nmn

m

aaa

253 33

Note: bases must be the same.

Power of a Quotient

Ex.

In general:

4

44

53

53

53

53

53

53

m

mm

ba

ba

Example 1 Evaluate Expressions with Negative Exponents

( ) 82 – – ( )42– Since bases are the same, add exponents

= ( ) 8 42 – – +

= ( ) 42 – –

1( )42–

= Move the power with negative exponent

=161

Evaluate power.

Example 2 Evaluate Quotients with Exponents

33

35 2Evaluate .

Since bases are the same, subtract exponents33

35 2

= ( )232

= 34

= 81 Evaluate power.

Checkpoint Evaluate Numerical Expressions

( )3221.

Evaluate the expression.

ANSWER 64

( )3502. ANSWER 1

3. ( ) 53 – –( )23– ANSWER271–

ANSWER2784.

32 3

Example 3 Simplify Algebraic Expressions

a.yx– 3

2Distribute the outside exponents

x 2

( )2y – 3 =

=x 2

y – 6 Simplify exponent.

= x 2y 6 Move the power with negative exponent

Example 3 Simplify Algebraic Expressions

= 25y y 5y – 6

Simplify exponent.

= 25y – 6 5 1 + + Add exponents of powers with same base.

= 25y 0

= 25 Apply Zero exponent prop

( )25y – 3 y 5y b. = ( )2y – 3 y 5y 52 Distribute outside exponent

Example 3 Simplify Algebraic Expressions

c.x 5y 2–

x 3y 6

=y 8

x 2

¿𝑥3 𝑦 6 𝑦2

𝑥5

To simplify the powers of x, think: which power has more x and by how many?

Move the power with negative exponent

The bottom x has more by 2.

Checkpoint Simplify Algebraic Expressions

( )32p p 45.8p 7

6.xy 4–

x 5y 3 x 4y 7

Simplify the expression.

7. ( )33b – 2 b 8 27b 2

–

sr 2

– 4

38.

1r 6s 12

ANSWER

Write an expression in simplified form for the Volume of the cylinder.

𝑉=𝜋 𝑟2 h r = radiush = height

Homework:

6.1 p.299 #8-34 even, 40, 42Disregard direction that says.

“Tell which properties you used”