exponents 07/24/12lntaylor ©. table of contents learning objectives bases exponents adding bases...
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Exponents
07/24/12 lntaylor ©
Table of Contents
Learning Objectives
Bases
Exponents
Adding Bases with exponents
Subtracting Bases with exponents
Multiplying Bases with exponents
Dividing Bases with exponents
Exponent of an exponent
Negative exponents
Fractional exponents
Explanation of a 0 exponent
07/24/12 lntaylor ©
LO1:
LO2:
Define and locate bases and exponents
Recognize bases and exponents which can be combined
LO3: Add and subtract bases and exponents
LO4: Multiply and divide bases and exponents
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Learning Objectives
LO5: Evaluate expressions with bases and exponents
Def1:
Def2:
Base – the number or variable whose exponents are expressed(4x3 where x is the base; 4 is the coefficient and 3 is the exponent)
Exponent – the number or variable to the upper right of the basewhich designates how many times the base is multiplied by itself(x3 = x*x*x; 3x4 = 3*x*x*x*x)
Def3: Coefficient – the number or letter in front of the base(3x4 where 3 is the coefficient)
Def4: Any base with an exponent of 0 always = 1! (x0 = 1; 120 = 1)
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Definitions
Bases
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3
Step 1
Note:
Look for the base
It can be a number
Note: It can be a variable (letter)
Note: It can be a combination of variables (letters)
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2x
2 x
2 3y3
Now you try
102 – 6x + mn
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10
Step 1
Note:
x
Look for the base
It can be a number
Note: It can be a variable (letter)
Note: It can be a combination of variables (letters)
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2+ mn– 6
Now you try
x2 – 7xy + 18
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x
Step 1
Note:
xy
Look for the base
It can be a number
Note: It can be a variable (letter)
Note: It can be a combination of variables (letters)
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2+ 18– 7
Exponents
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3
Step 1
Note:
Look for the exponent
It can be a number
Note: It can be a variable (letter)
Note: There can be more than one exponent
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2x
y3 x
2 my
Note: Everything gets an exponent!!!!!
1
Now you try
x2 + 10xn + 18y
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x
Step 1
Note:
Look for the exponent
It can be a number
Note: It can be a variable (letter)
Note: There can be more than one exponent
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2+ 10 x
n+ 18 y
Note: Everything gets an exponent!!!!!
1 11
Now you try
12x2 + 14ym – 10
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12 x
Step 1
Note:
Look for the exponent
It can be a number
Note: It can be a variable (letter)
Note: There can be more than one exponent
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2+ 14 y - 10
Note: Everything gets an exponent!!!!!
11 1 m
Adding Bases with exponents
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x
Step 1
Step 2
Look at each variable and its exponent
Combine the variables and exponents that are exactly the same
Step 3 Rewrite the expression
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2x - 10x
212 + 2
14x2
Now you try
10x2 + 12x2 – x
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x
Look at each variable and its exponent
Combine the variables and exponents that are exactly the same
Rewrite the expression
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2x - x
210 + 12
22x2
Step 1
Step 2
Step 3
Now you try
10xy2 + 12xy2 – x2
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xy
Step 1
Step 2
Look at each variable and its exponent
Combine the variables and exponents that are exactly the same
Step 3 Rewrite the expression
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2xy - x2
210 + 12
22xy2
Subtracting Bases with exponents
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x
Step 1
Step 2
Look at each variable and its exponent
Combine the variables and exponents that are exactly the same
Step 3 Rewrite the expression
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2x - 10x
212 - 2
10x2
Now you try
10x2 - 12x2 – x
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x
Step 1
Step 2
Look at each variable and its exponent
Combine the variables and exponents that are exactly the same
Step 3 Rewrite the expression
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2x - x
210 - 12
- 2x2
Now you try
12xy2 – 12xy2 – x2
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xy
Step 1
Step 2
Look at each variable and its exponent
Combine the variables and exponents that are exactly the same
Step 3 Rewrite the expression
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2xy - x2
212 - 12
0
Multiplying Bases with exponents
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x
Step 1:
Step 2:
Multiply the coefficients (here all the coefficients are = 1)
If the bases are identical, write it downIf the bases are different, write them down
Step 3: Add the exponents of similar bases
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2(x )
3= x x
2(xy )
3= x y
5 3 31
Now you try
2x2 (6x7)
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x
Step 1:
Step 2:
Multiply the coefficients
If the bases are identical, write it downIf the bases are different, write them down
Step 3: Add the exponents of similar bases
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2( x )
7 x
92 6 12=
Now you try
12xy2 (6x7y2z)
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xy
Step 1:
Step 2:
Multiply the coefficients
If the base is identical, write it downIf the bases are different, write them down
Step 3: Add the exponents of similar bases
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2( x )
7 x y z
812 6 72=y z
2 41
Now you try
¾xy2 (4x5y4z)
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xy
Step 1:
Step 2:
Multiply the coefficients
If the bases are identical, write it downIf the bases are different, write them down
Step 3: Add the exponents of similar bases
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2( x )
5 x y z
6¾ 4 3=y z
4 61
Dividing Bases with exponents
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x
Step 1:
Step 2:
Divide the coefficients
Determine where the bases go (numerator or denominator)by looking for the largest exponents
Step 3: Subtract the exponents of similar bases
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7
x2 x
512
6 2=
Note: Exponent answers are always positiveDetermine if the exponent is in the numerator or denominator
Now you try
xy2
4x5y4z
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xy
Step 1:
Step 2:
Divide the coefficients
Determine where the bases go (numerator or denominator)by looking for the largest exponents
Step 3: Subtract the exponents of similar bases
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2
x
5 x y z
44
1=
y z4 2
1
Note: Final exponent answers are always positiveDouble check your work
4
Now you try
6x9y2z6
12x7y12z6
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x y
Step 1:
Step 2:
Divide the coefficients
Determine where the bases go (numerator or denominator)by looking for the largest exponentsNote here that z6 / z6 = 1 so there is no need to write z
Step 3: Subtract the exponents of similar bases
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2
x
7
x
y
2
12
1=
y z12 10
9
Note: Final exponent answers are always positiveDouble check your work and rewrite if necessary
2
6 z6
6
= x2
2y10
Now you try
9x7y2z6
12x7y2z6
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x y
Step 1:
Step 2:
Divide the coefficients
Determine where the bases go (numerator or denominator)by looking for the largest exponents
Step 3: Subtract the exponents of similar bases
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2
x
712
3=
y z2
7
Note: Final exponent answers are always positiveDouble check your work and rewrite if necessary
4
9 z6
6
Exponent of an exponent
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Step 1:
Step 2:
Look for the exponent next to a ( ); (2x)5
Everything gets an exponent (no need here)
Step 3: Rewrite all bases and multiply the exponents
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(x )7
x14
=
Step 4: Simplify if necessary
2
Now you try
( 2x7y2z )3
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Step 1:
Step 2:
Look for the exponent next to a ( )
Everything gets an exponent
Step 3: Rewrite all bases and multiply the exponentsBe patient and let the computer work!!!
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( )7
23(1)
=
Step 4: Simplify
22 x y z3 3(7) 3(2) 3(1)
11 x y z = 23x
21y
6z
3= 8x
21y
6z
3
Now you try
( 2x4y5z )7
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Step 1:
Step 2:
Look for the exponent next to a ( )
Everything gets an exponent
Step 3: Rewrite all bases and multiply the exponentsBe patient and let the computer work!!!
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( )4
27(1)
=
Step 4: Simplify
52 x y z7 7(4) 7(5) 7(1)
11 x y z = 27x
28y
35z
7= 128x
28y
35z
7
Negative exponents
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Step 1:
Step 2:
Look for negative exponents
Everything gets an exponent (no need here)
Step 3: Remember that everything is a fractionFlip over only the base with the negative exponentRemove the negative ( - ) sign
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x- 7
17
=
Step 4: Simplify if necessary
1 x
Now you try
2x4y5z -7
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Step 1:
Step 2:
Look for negative exponents
Everything gets an exponent
Step 3: Remember that everything is a fractionFlip over only the base with the negative exponentMake sure you removed the negative ( - ) sign
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2 x4y
5z
- 7
21 x
4y
5
7=
Step 4: Simplify if necessary
1 z
1
Now you try
2x4y5z -7
x-8yz
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Step 1:
Step 2:
Look for negative exponents
Everything gets an exponent
Step 3: Since you are dividing you can subtract the negative exponent(x4 / x-8 is x 4- - 8 or x 12 )Make sure you removed the negative ( - ) sign
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2 x4y
5z
- 7
21 x
12y
4
8
=
Step 4: Simplify if necessary
x-8y z z
1
1 1
2x12
y4
8
=
z
Fractional Exponents
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4
Step 1
Note:
Look for the exponent
It can be sometimes be a fraction (rational number)
Step 2: The base and numerator go under the radical (square root symbol)
Step 3: The denominator goes outside the radical
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1
2
√ 412
Note: An exponent of ½ means square root; 1/3 means cube root
= 2
Now you try
81/3
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8
Step 1
Note:
Look for the exponent
It can be sometimes be a fraction (rational number)
Step 2: The base and numerator go under the radical (square root symbol)
Step 3: The denominator goes outside the radical
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1/3
√ 813
Note: An exponent of ½ means square root; 1/3 means cube root
= 2
Now you try
82/3
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8
Step 1
Note:
Look for the exponent
It can be sometimes be a fraction (rational number)
Step 2: The base and numerator go under the radical (square root symbol)
Step 3: The denominator goes outside the radical
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2/3
√ 823
Note: An exponent of ½ means square root; 1/3 means cube root
= √643 = 4
Exponent = 0
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3 = 1
Step 1
Note:
Proof of 30 = 1
Use 3 as the base and any exponent you wish – we will use 8Divide identical bases and exponents
Note: Anything divided by itself is 1 (the exception is of course base = 0)
Note: Dividing identical bases and exponents requires subtraction
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0
38
38
= 3 8-8= 1 = 3
0
Note: This leaves an exponent of 0The only way you get an exponent of 0 is to divide something by itselfTherefore proving an exponent of 0 = 1
= 1
Now you try
x0 = 1
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x = 1
Step 1
Note:
Proof of x0 = 1
Use x as the base and any exponent you wish – we will use 10Divide identical bases and exponents
Note: Anything divided by itself is 1 (the exception is of course base = 0)
Note: Dividing identical bases and exponents requires subtraction
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0
x10
x10
= x 10-10= 1 = x
0
Note: This leaves an exponent of 0The only way you get an exponent of 0 is to divide something by itselfTherefore proving an exponent of 0 = 1
= 1
Now you try
7x0y
0 = 7
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Step 1
Note:
Proof of 7x0y0 = 7
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x0 0y7 = 7
7 (1)(1) = 7
Note: 7 has an exponent of 1
1
Anything with a 0 exponent = 1 (except base 0 of course)
Note: This expression = 7