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Exponent PowerPoint - Clackamas Middle College

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RATIONAL EXPONENTS
Basic
terminology
Substitution
and
evaluating
Laws of
Exponents
Assignments
Multiplication
Properties
•Product of powers
•Power to a power
•Power of a product
Assignments
Zero and negative
exponents
Division properties
of exponents
•Quotient of powers
•Power of a quotient
Basic Terminology
Exponent пЂ­ In Exponentia l notation , the number of times the base
is used as a factor .
Base пЂ­ In Exponentia l notation , the number or Variable
MULTIPLICA
2
that undergoes
repeated
TION .
4
EXPONENT
means
2 п‚ґ 2 п‚ґ 2 п‚ґ 2 пЂЅ 16
BASE
Important Examples
IMPORTANT EXAMPLES
пЂ­ 3 means
4
( пЂ­ 3 ) means
4
пЂ­ 3 means
3
пЂ­ ( 3 п‚ґ 3 п‚ґ 3 п‚ґ 3 ) пЂЅ пЂ­ 81
( пЂ­ 3 ) п‚ґ ( пЂ­ 3 ) п‚ґ ( пЂ­ 3 ) п‚ґ ( пЂ­ 3 ) пЂЅ 81
пЂ­ ( 3 п‚ґ 3 п‚ґ 3 ) пЂЅ пЂ­ 27
( пЂ­ 3 ) means ( пЂ­ 3 ) п‚ґ ( пЂ­ 3 ) п‚ґ ( пЂ­ 3 ) пЂЅ пЂ­ 27
3
Variable Examples
Variable Expressions
x
5
3
means ( use parenthese s for multiplica tion ) ( x )( x )( x )( x )( x )
y means ( y )( y )( y )
Substitution and Evaluating
STEPS
1. Write out the original problem.
2. Show the substitution with parentheses.
3. Work out the problem.
Example : Solve if x пЂЅ 4 ;
Solve if x пЂЅ 4 ;
(4)
3
x
x
3
3
= 64
More Examples
Evaluate the variable expression when x = 1, y = 2, and w = -3
( x) пЂ« ( y)
2
2
(x пЂ« y)
Step 1
( x) пЂ« ( y)
2
2
2
2
wx
y
Step 1
Step 1
(x пЂ« y)
Step 2
(1) пЂ« ( 2 )
2
2
wx
y
Step 2
Step 2
пЂЁ(1) пЂ« ( 2 ) пЂ©
2
Step 3
Step 3
(пЂ­ 3 )(1)
2
Step 3
1пЂ« 4 пЂЅ 5
(3) пЂЅ 9
2
( пЂ­ 3 )( 1) пЂЅ пЂ­ 3
MULTIPLICATION PROPERTIES
PRODUCT OF POWERS
This property is used to combine 2 or more exponential expressions with the SAME base.
2 п‚ґ2
3
3
4
( x )( x )
5
( 2 п‚ґ 2 п‚ґ 2 )( 2 п‚ґ 2 п‚ґ 2 п‚ґ 2 п‚ґ 2 )
пЂЁ( x )( x )( x ) пЂ© пЂЁ( x )( x )( x )( x ) пЂ©
2
x
7
8
256
MULTIPLICATION PROPERTIES
POWER TO A POWER
This property is used to write and exponential expression as a single power of the base.
2
(5 )
2
(x )
4
3
2
2
2
5
( 5 )( 5 )( 5 )
2
2
2
2
( x )( x )( x )( x )
6
x
8
MULTIPLICATION PROPERTIES
POWER OF PRODUCT
This property combines the first 2 multiplication properties to simplify exponential expressions.
( пЂ­ 6 п‚ґ 5)
( 5 xy )
3
(4 x ) п‚· x
2
3
2
( пЂ­ 6 ) п‚ґ (5 )
2
3
3
2
3
125 x y
3
( 5 )( x )( y )
( 4 )( x ) п‚· x
3
5
( 64 )( x ) п‚· x
6
5
2
3
36 п‚ґ 25 пЂЅ 900
( 64 ) пЂЁ( x )( x )( x ) пЂ© п‚· x
5
64 x
3
2
11
2
2
5
MULTIPLICATION PROPERTIES
SUMMARY
PRODUCT OF POWERS
x п‚·x пЂЅ x
a
b
aпЂ«b
ADD THE EXPONENTS
POWER TO A POWER
пЂЁx пЂ©
a b
пЂЅ x
a п‚·b
MULTIPLY THE EXPONENTS
POWER OF PRODUCT
( xy ) пЂЅ x y
a
a
a
ZERO AND NEGATIVE EXPONENTS
ANYTHING TO THE ZERO POWER IS 1.
3 пЂЅ 27
2
пѓ¦ 1 пѓ¶
пЂЅ 2пѓ§ 2 пѓ· пЂЅ 2
пѓЁx пѓё x
3
2x
3 пЂЅ9
2
пЂ­2
3 пЂЅ3
1
3 пЂЅ1
(2 x)
0
3
пЂ­1
пЂЅ
1
3
3
пЂ­2
пЂЅ
1
1
3
3
пЂ­3
пЂЅ
2
1
3
3
пЂЅ
пЂ­2
пЂЅ
1
(2 x)
1
2
1
пЂЅ
2
2 x
2
пЂЅ
1
4x
3
пЂЅ
1
9
пЂЅ
1
27
1
3
пЂ­4
пЂЅ
1
1
3
4
пЂЅ 1п‚ё
1
3
4
пЂЅ 1п‚ґ
3
4
1
пЂЅ 3 пЂЅ 81
4
2
DIVISION PROPERTIES
QUOTIENT OF POWERS
This property is used when dividing two or more exponential expressions with the same base.
x
5
x
3
пЂЅ
( x )( x )( x )( x )( x )
( x )( x )( x )
пЂЅ
( x )( x )
пЂЅ x
1
1
x
пЂ­3
x
4
1
1
1
1
4
x
пЂЅ 4 пЂЅ 3 п‚ёx пЂЅ 3п‚ґ 4 пЂЅ 7
x
x
x
x
x
3
2
DIVISION PROPERTIES
POWER OF A QUOTIENT
4
2 4
8
пѓ¦x пѓ¶
(x )
x
пѓ§ 3пѓ· пЂЅ
пЂЅ 12
3 4
пѓ§ y пѓ·
(y )
y
пѓЁ
пѓё
2
Hard Example
пЂ­2
пѓ¦ 2 xy
пѓ§ 3 пЂ­4
пѓ§ 3x y
пѓЁ
3
8x y
9
3
3 3 пЂ­6
пЂ­2 3
3 12
пѓ¶
2 x y
( 2 xy )
8x y
пѓ· пЂЅ
пЂЅ 3 9 пЂ­12 пЂЅ
пЂЅ
пѓ·
3 пЂ­4 3
9 6
3 x y
(3 x y )
27 x y
пѓё
12
27 x y
6
пЂЅ
8y
6
27 x
6
Summary of Zero, Negative, and
Division Properties
ZERO, NEGATIVE, AND DIVISION
PROPERTIES
( x) пЂЅ 1
0
Zero power
Negative Exponents
пЂ­a
x
пЂЅ
1
x
Quotient of powers
x
a
x
b
пЂЅ x
a пЂ­b
a
Power of a quotient
and
1
x
пЂ­a
a
пЂЅ x
a
пѓ¦xпѓ¶
x
пѓ§пѓ§ пѓ·пѓ· пЂЅ a
y
пѓЁ yпѓё
a
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