Properties of Exponents

Home , Fifth power (algebra)

1 Algebra II

2015-11-09

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2 Table of Contents

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Review of Integer Exponents Fractional Exponents Exponents with Multiple Terms

Identifying Like Terms

Evaluating Exponents Using a Calculator Standards

3 Review of Integer Exponents

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4 Powers of Integers

Just as multiplication is repeated addition, an exponent represents repeated multiplication.

For example, a5 read as "a to the fifth power" = a · a · a · a · a

In this case, a is the base and 5 is the exponent.

The base, a, is multiplied by itself 5 times.

5 Powers of Integers

Make sure when you are evaluating exponents of negative numbers, you keep in mind the meaning of the exponent and the rules of multiplication.

For example, , which is the same as .

However, Notice the difference!

Similarly, but .

6 1 Evaluate:

64 Answer

7 2 Evaluate:

-128 Answer

8 3 Evaluate:

81 Answer

9 Properties of Exponents

The properties of exponents follow directly from expanding them to look at the repeated multiplication they represent.

Work to understand the process by which we find these properties and if you can't recall what to do, just repeat these steps to confirm the property.

We'll use 3 as the base in our examples, but the properties hold for any base. We show that with base a and powers b and c.

We'll use the facts that:

10 Properties of Exponents

We need to develop all the properties of exponents so we can discover one of the inverse operations of raising a number to a power which is finding the root of a number.

This concept will emerge from the final property we'll explore.

But, getting to that property requires understanding the others first.

11 Multiplying with Exponents

When multiplying numbers with the same base, add the exponents. M a t h P r c i e

12 Dividing with Exponents When dividing numbers with the same base, subtract the exponent of the denominator from that of the numerator.

13 4 Simplify:

B C C Answer D

14 5 Simplify:

B B

C Answer D

15 6 Simplify:

B C Answer C

16 7 Simplify:

B C Answer C

17 8 Simplify:

B A Answer C

18 9 Simplify:

B B Answer C

19 10 Simplify:

A D B

Answer

20 An Exponent of Zero Any base raised to the power of zero is equal to 1.

Based on the property for multiplication:

We know that any number times 1 is equal to itself, thus

21 11 Evaluate:

1 Answer

22 12 Evaluate:

2 Answer

23 13 Evaluate:

7 Answer

24 Negative Exponents A negative exponent moves the number from the numerator to denominator, and vice versa.

Based on the multiplication and zero exponent properties:

Dividing by 31, we see that:

25 Negative Exponents

By definition:

26 14 Rewrite an equivalent to the following using positive exponents:

A Answer B

27 15 Rewrite an equivalent to the following using positive exponents:

B A Answer

28 16 Rewrite an equivalent to the following using positive exponents:

B C Answer

29 17 Rewrite an equivalent to the following using positive exponents:

A D

B Answer

30 18 Which expression is equivalent to ?

B A Answer C

From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

31 19 What is the value of ?

A B

B Answer

From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

32 20 Which expression is equivalent to ?

C C Answer D

From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

33 21 Which expressions are equivalent to ? Select all that apply.

B B & E Answer C

From PARCC EOY sample test non-calculator #13

34 22 Which expressions are equivalent to ? Select all that apply.

B B, D & E Answer C

From PARCC EOY sample test non-calculator #13

35 Raising Exponents to Higher Powers

When raising a number with an exponent to a power, multiply the exponents.

36 Raising Exponents to Higher Powers

It's important to note that when you have multiple numbers and variables being raised to a higher power, the exponent outside gets multiplied by EACH exponent inside.

For example,

37 23 Simplify

A B D Answer C

38 24 Simplify

B Answer C

39 25 Simplify

B D

C Answer

40 26 Simplify:

A B B

C Answer D

41 27 When is divided by , and , the quotient is

A r e

B w s n

C A D

From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

42 28 If and , what is the value of ?

A r e

B w s n

C A D

From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

43 29 The expression is equivalent to:

A B B C Answer D

From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

44 30 The expression is equivalent to:

A B D C Answer D

From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

45 Fractional Exponents

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46 Exponents & Roots

We already know that roots can "undo" an exponent, and exponents can "undo" roots. For example,

and

Next, we will be introduced to a way to "undo" exponents using other exponents. In other words, we will find exponents which will operate the same as roots.

47 Roots as Exponents

The square root of a number is equivalent to taking that number to the 1/2 exponent. To prove this we will begin by considering the multiplication property for exponents,

And by definition, Comparing the 2 equations, we see that:

And furthermore,

48 Special Reciprocal Exponents

Similarly, raising a number to the power 1/3 is the same as taking the cube root of the number.

And in a general, a number to the power 1/n is the same as taking the nth root of the number. M a t h P r c i e

The inverse of raising a number to a power is raising it to the reciprocal power.

49 Reciprocal Exponents

So raising a number to the power of 1/n is the same as taking the nth root of the number.

If you raise a number to n, then take the nth root, you get back where you started...the original number.

It's the same as taking a number and adding and then subtracting the same amount.

Or, multiplying and then dividing by the same number.

The root is an inverse of raising to a power.

And, raising to the power 1/n is the inverse of raising to n.

50 31 Evaluate: r e w s n A

51 32 Evaluate: r e w s n A

52 33 Evaluate: r e w s n A

53 34 Evaluate: r e w s n A

54 35 Evaluate: r e w s n A

55 36 Evaluate: r e w s n A

56 37 Evaluate: r e w s n A

57 38 Evaluate: r e w s n A

58 39 Evaluate: r e w s n A

59 40 Evaluate: r e w s n A

60 41 Evaluate: r e w s n A

61 42 Evaluate: r e w s n A

62 43 Evaluate: r e w s n A

63 44 Evaluate: r e w s n A

64 45 Evaluate: r e w s n A

65 Fractional Exponents with Numerators ≠ 1

A fractional exponent is performing two different operations.

The numerator raises the base to a power, and the denominator takes the root of the base.

In this case, the numerator raises a to the power m while the denominator takes the nth root.

The order they are applied does not matter. They are commutative.

66 Power Over Root s e t o N

r e h c

solar POWER a e T

tree ROOTS

67 Exponents with Numerators ≠ 1

While order doesn't affect the result, it's often easier to do the root first, so the number is smaller.

For instance, let's evaluate (81)5/4, both ways.

(815)1/4 (811/4)5 M a t h P r c i e (3,486,784,401)1/4 (3)5

243 243

The result is the same, but the solution on the left would be very difficult without a calculator, while that on the right is pretty easy.

Generally, taking the root first is easier.

Also, there's one other property worth noting. Can you find a shortcut? How would your shortcut make the problem easier?

68 Exponents with Numerators ≠ 1

(81)5/4 By the property of the multiplication of exponents (81)4/4 (81)1/4 Since 4/4 = 1 (81)1 (81)1/4 Any number raised to the power 1 is itself (81)(81)1/4 The 4th root of 81 is 3 (81)(3) Simple multiplication 243

So, if the exponent is an improper fraction, you can save time by just separating out the numerator into an integer and the fractional remainder.

69 Teachers:

Use the questions located on the pull tab for the next 9 slides. M a t h P r c i e

70 46 Evaluate: r e w s n A

71 47 Evaluate: r e w s n A

72 48 Evaluate: r e w s n A

73 49 Evaluate: r e w s n A

74 50 Evaluate: r e w s n A

75 51 Which of the below is equivalent to:

A r

B e w s n C A

76 52 Which of the below is equivalent to:

A r e w s

B A

77 53 Which of the below is equivalent to:

A r e w s

B A

78 54 Which expression is equivalent to ?

A r

B e w s n

C A

PARCC PBA Algebra II Sample Item - Non-calculator - #7

79 Fractional Exponents with Expressions

Aside from having one number or one variable being expressed with fractional exponents, you can have multiple terms inside of the radical or raised to a fractional exponent.

For example

80 Fractional Exponents with Expressions

When given an expression raised to a fractional exponent, it's best to start from the inside and work your way out. Write the expression from the innermost root as the expression raised to that power. Then continue this pattern as you work your way out. Finally, use the rules of exponents to simplify.

For example M a t h P r c i e

81 55 If

for x ≥ 0, where b is a constant, what is the value of b?

A e w s n B A

82 56 If

for x ≥ -5/4, where d is a constant, what is the value of d?

A r e

w B s n A C

83 57 If

for x ≥ 4, where c is a constant, what is the value of c? r e w s n A

84 58 If

for x ≥ -3, where g is a constant, what is the value of g? r e w s n A

85 59 If

for x ≥ -1, where a is a constant, what is the value of a? r e w s n A

Algebra II - EOY - Calculator Section - #5

86 Exponents With Multiple Terms

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87 Exponents with More Than One Term

An exponent may have more than one term such as .

By definition terms are either added to, or subtracted from, one another.

The multiplication properties of exponents allow us to translate the addition or subtraction of exponents into multiplication or division of terms.

Or, reversing the direction:

88 Exponents with More Than One Term

A base raised to an exponent with n terms can be treated as a multiplication of n bases raised to each of the terms on the original exponent.

This is just the multiplication property of exponents, done in the opposite direction.

89 Exponents with More Than One Term

How would you find an equivalent expression for this such that the exponent in the new expression has only one term? M a t h P r c i e Discuss.

90 Exponents with More Than One Term

For example, find an equivalent expression for this:

This is as far as we can proceed. We cannot multiply 64 and 4 since 4 is raised to the power of x, while 64 is not.

Whether this form is more or less useful depends on the problem.

91 Exponents with More Than One Term

For example, find an equivalent expression for this:

r e w s n A

92 60 Which of the below is equivalent to:

A r

B w s n C A

93 61 Which of the below is equivalent to:

A r e w

B n A C

94 62 Which of the below is equivalent to:

A r e

B w s n A C

95 63 Which of the below is equivalent to:

A r e

B w s n C A

96 Simplifying Expressions

Expressions with unlike terms can sometimes be simplified to a single term through this process.

How could you simplify this expression to one term?

M a t h P r c i e

97 Simplifying Expressions

First, simplify the second term

Then combine terms

98 64 Which of the below is equivalent to:

A r

B e w s n C A

99 65 Which of the below is equivalent to:

A r e w

B s n A C

100 66 Which of the below is equivalent to:

A r e

w B s n A C

101 67 Consider the expression . Part A Which is an equivalent form of the given expression?

A r e w s n

B A

PARCC EOY Algebra II - Calculator #23

102 68 Consider the expression . Part B This expression can also be rewritten in the form of where a is a constant. What is the value of a? r

A e w s n

B A

PARCC EOY Algebra II - Calculator #23

103 Identifying Like Terms

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104 Like Terms Sometimes the properties of exponents can be used to transform terms so they becomes like, and can then be combined.

For instance, it could appear that the expression

has no like terms and cannot be simplified

and have different bases and seem unlike.

But can be written as which is equal to , yielding:

This cannot be further simplified, since 2 is not raised to x.

105 Exponents with More Than One Term

For example, simplify this expression:

106 69 Which of the below is equivalent to:

A r e w B s n A C

107 70 Which of the below is equivalent to:

A r e

B w s n A C

108 71 Which of the below is equivalent to:

A r e w s

B n A

109 Solving Equations Using the Properties of Exponents

The properties of exponents can be used to transform terms on one or both sides of an equation to find the value(s) of the variable.

For instance, it could appear that the equation M a t h P r c i e has no like terms and cannot be solved because both terms have different bases and seem unlike. But 9 can be written as 32, yielding:

110 Solving Equations Using the Properties of Exponents

Now, since our bases are the same, we set the exponents equal and write the equation to solve for x.

111 Solving Equations Using the

Properties of Exponents It appears that this equation has no like terms and cannot be simplified because Let's try another example: all three terms have different bases and seem unlike. r

e But can be written as which is

w equal to , yielding: s n

A Since the bases are the same, we can write an equation and solve for x.

112 Solving Equations Using the Properties of Exponents

Another example: r e w s n A

113 Solving Equations Using the Properties of Exponents

One last example: r e w s n A

114 72 Consider the equation Which is an equivalent equation?

A e w s n

B A

115 73 Consider the equation

What are the values of x? Select all that apply.

A r e w

B s n A C -1

D 1 E F

116 74 Consider the equation Which is an equivalent equation?

A r e w s

B A

117 75 Consider the equation

What are the values of x? Select all that apply.

A r e

B w s n A C

E F

118 76 Consider the equation Which is an equivalent equation?

A r e w s

B n A

119 77 Consider the equation

What are the values of x? Select all that apply.

A -4 r e

B -2 w s n C -1 A

D 1 E 2 F 4

120 78 Consider the equation Which is an equivalent equation?

A r e w

s B n A C

121 79 Consider the equation

What are the values of x? Select all that apply.

A -3 r

B -1.5 e w s C -1 n A

D 1 E 1.5 F 3

122 80 Solve the equation for x. r e w s n A

PARCC PBA Algebra II - Non-Calculator #2

123 81 Consider the equation

Part A Which equation is equivalent to the equation shown? r e w s n

A A B

C D

PARCC EOY Algebra II - Non-Calculator #4

124 82 Consider the equation

Part B Which values are solutions to the equation? Select all that apply. r

A -2 e w s n

B -1 A

C - 1 2 D 1 2 E 1

F 2

PARCC EOY Algebra II - Non-Calculator #4

125 Evaluating Exponents Using a Calculator

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126 Evaluating Exponents Using a Calculator

There are different ways to evaluate exponents when using your calculator.

The first way is using the root command.

If we have the example , we can start by entering the root that we are taking, in our case 5.

127 Evaluating Exponents Using a Calculator

Example:

Next, bring up the menu of different math operations by pressing the "MATH" button.

128 Evaluating Exponents Using a Calculator

Example:

x Scroll down to the 5th option √ and press "Enter" Note: Or, you could press "5" instead of scrolling

129 Evaluating Exponents Using a Calculator

Example:

Then, type in the number 32, putting parentheses around it.

Getting into a habit of including parentheses when taking a root is beneficial, especially when the expression involves more operations.

130 Evaluating Exponents Using a Calculator

Example:

Last, push "ENTER" and your answer will be calculated.

Our answer is 2.

131 83 Use your calculator to evaluate the number. Round your answer to the 4th decimal place. r e w s n A

Use a calculator

132 84 Use your calculator to evaluate the number. Round your answer to the 4th decimal place. r e w s n A

Use a calculator

133 85 Use your calculator to evaluate the expression. Round your answer to the 4th decimal place. r e w s n A

Use a calculator

134 Evaluating Exponents Using a Calculator

There are different ways to evaluate exponents when using your calculator.

The second way is using the exponent notation.

We can rewrite our example ,

as .

To start, enter the 32 into your calculator.

135 Evaluating Exponents Using a Calculator

Example: =

Next, use the "^" button, located directly above the division sign, to raise 32 to a power.

136 Evaluating Exponents Using a Calculator

Example: =

Then, enter the exponent of 1/5 into the calculator. Make sure that the exponent is in parentheses. Otherwise the calculator will evaluate 32 to the 1st power and then divide by 5.

137 Evaluating Exponents Using a Calculator

Example: =

Last, push "ENTER" and your answer will be calculated. Our answer is 2.

138 86 Evaluate: Round your answer to 4 decimal places. r e w s n A

Use a calculator

139 87 Evaluate: Round your answer to 4 decimal places. r e w s n A

Use a calculator

140 88 Evaluate: Round your answer to 4 decimal places. r e w s n A

Use a calculator

141 89 Evaluate: Round your answer to 4 decimal places. r e w s n A

Use a calculator

142 Standards

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143 Throughout this unit, the Standards for Mathematical Practice are used. MP1: Making sense of problems & persevere in solving them. MP2: Reason abstractly & quantitatively. MP3: Construct viable arguments and critique the reasoning of others. MP4: Model with mathematics. MP5: Use appropriate tools strategically. MP6: Attend to precision.

MP7: Look for & make use of structure. M a t h P r c i e MP8: Look for & express regularity in repeated reasoning. Additional questions are included on the slides using the "Math Practice" Pull-tabs (e.g. a blank one is shown to the right on this slide) with a reference to the standards used. If questions already exist on a slide, then the specific MPs that the questions address are listed in the Pull-tab.

144