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”