exponents 07/24/12lntaylor ©. table of contents learning objectives bases exponents adding bases...

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

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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