8th Grade Math

Numerical Roots & Radicals

www.njctl.org

2012-12-03

Numerical Roots and Radicals

Approximating Square Roots

Rational & Irrational Numbers

Radical Expressions Containing Variables

Simplifying Non-Perfect Square Radicands

Simplifying Perfect Square Radical Expressions

Simplifying Roots of Variables

Click on topic to go to that

section.

Squares, Square Roots & Perfect Squares

Squares of Numbers Greater than 20

Properties of Exponents

Solving Equations with Perfect Square & Cube Roots

Common Core Standards: 8.NS.1-2; 8.EE.1-2

Squares, Square Roots and Perfect Squares

Return to Table of Contents

Area of a Square

The area of a figure is the number of square units needed to cover the figure.

The area of the square below is 16 square units because 16 square units are needed to COVER the figure...

A = 42 = 4 4

= 16 sq.units

Area of a Square

The area (A) of a square can be found by squaring its side length, as shown below:

Click to see if the answer found with the Area formula is

correct!

A = s2

4 units

The area (A) of a square is labeled as

square units, or units2, because you

cover the figure with squares...

1 What is the area of a square with sides of 5 inches?

A

B

D

C

16 in2

25 in2

20 in2

30 in2

2 What is the area of a square with sides of 6 inches?

16 in2

24 in2

20 in2

36 in2

A

B

D

C

3 If a square has an area of 9 ft2, what is the

length of a side?

A

B

D

C

2 ft

3 ft

2.25 ft

4.5 ft

4 What is the area of a square with a side length of 16 in?

5 What is the side length of a square with an area of 196 square feet?

When you square a number you multiply it by itself.

52 = 5 5 = 25 so the square of 5 is 25.

You can indicate squaring a number with an exponent of 2, by asking for the square of a number, or by asking for a number squared.

What is the square of seven?

What is nine squared?

49

81

Make a list of the numbers 1-15 and then square each of them.

Your paper should be set up as follows:

Number Square 1 1 2 4 3

(and so on)

Number Square

1 1 2 4 3 9 4 16 5 25 6 36 7 49 8 64 9 81 10 100 11 121 12 144 13 169 14 196 15 225

The numbers in the right column are squares of the numbers in the left column.

If you want to "undo" squaring a number, you must take the square root of the number.

So, the numbers in the left column are the square roots of the numbers in the right column.

Square Root Square

1 12 43 94 165 256 367 498 649 8110 10011 12112 14413 16914 19615 225

The square root of a number is found by undoing the squaring. The symbol for square root is called a radical sign and it looks like this:

Using our list, to find the square root of a number, you find the number in the right hand column and look to the left.

So, the 81 = 9

What is 169?

Square Perfect Root Square

1 12 43 94 165 256 367 498 649 8110 10011 12112 14413 16914 19615 225

When the square root of a number is a whole number, the number is called a perfect square.

Since all of the numbers in the right hand column have whole numbers for their square roots, this is a list of the first 15 perfect squares.

Find the following. You may refer to your chart if you need to.

6 What is ?1

7 What is ?81

8 What is the square of 15 ?

9 What is ?256

10 What is 132?

11 What is ?196

12 What is the square of 18?

13 What is 11 squared?

14 What is 20 squared?

Squares of Numbers Greater than 20

Return to Table of Contents

Think about this...

What about larger numbers?

How do you find ?

It helps to know the squares of larger numbers such as the multiples of tens.

102 = 100

202 = 400

302 = 900

402 = 1600

502 = 2500

602 = 3600

702 = 4900

802 = 6400

902 = 8100

1002 = 10000

What pattern do you notice?

For larger numbers, determine between which two multiples of ten the number lies.

102 = 100 1

2 = 1

202 = 400 2

2 = 4

302 = 900 3

2 = 9

402 = 1600 4

2 = 16

502 = 2500 5

2 = 25

602 = 3600 6

2 = 36

702 = 4900 7

2 = 49

802 = 6400 8

2 = 64

902 = 8100 9

2 = 81

1002 = 10000 10

2 = 100

Next, look at the ones digit to determine the ones digit of your square root.

2809

Examples:

Lies between 2500 & 3600 (50 and 60)Ends in nine so square root ends in 3

or 7Try 53 then 57

532 = 2809

Lies between 6400 and 8100 (80 and 90)

Ends in 4 so square root ends in 2 or 8Try 82 then 88

822 = 6724 NO!

882 = 7744

7744

15 Find.

16 Find.

17 Find.

18 Find.

19 Find.

20 Find.

21 Find.

22 Find.

23 Find.

Simplifying Perfect Square Radical Expressions

Return to Table of Contents

Can you recall the perfect squares from 1 to 400?

12 =

8

2 = 15

2 =

22 = 9

2 = 16

2 =

32 = 10

2 = 17

2 =

42 = 11

2 = 18

2 =

52 = 12

2 = 19

2 =

62 = 13

2 = 20

2 =

72 = 14

2 =

Square Root Of A Number

Recall: If b2 = a, then b is a square root of a.

Example: If 42 = 16, then 4 is a square root of 16

What is a square root of 25? 64? 100?

5 8 10

Square Root Of A Number

Square roots are written with a radical symbol

Positive square root: = 4

Negative square root:- = - 4

Positive & negative square roots: = 4

Negative numbers have no real square roots no real roots because there is no real number that, when squared, would equal -16.

Is there a difference between

Which expression has no real roots?

&

Evaluate the expressions:

?

is not real

Evaluate the expression

24

?25

= ?26

27

28

= ?29

A

B

C

3

No real roots

-3

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.

A

B

D

C

-10

16

64

4

30 The expression equal to is equivalent to a positive integer when b is

Square Roots of Fractions

ab

= b 0

1649

= = 4

7

Try These

31

A

B D

C

no real solution

32

A

B D

C

no real solution

33

A

B D

C

no real solution

A

B D

C

no real solution

34

A

B D

C

no real solution

35

Square Roots of Decimals

Recall:

To find the square root of a decimal, convert the decimal to a fraction first. Follow your steps for square roots of fractions.

= .05

= .2

= .3

Evaluate 36

A

B D

C

no real solution

Evaluate 37

A

C D

B.06 .6

6 No Real Solution

Evaluate 38

A

C D

B.11 11

1.1 No Real Solution

Evaluate 39

A

C D

B.8 .08

No Real Solution

Evaluate 40

A

C D

B

No Real Solution

Approximating Square Roots

Return to Table of Contents

All of the examples so far have been from perfect squares.

What does it mean to be a perfect square?

The square of an integer is a perfect square.A perfect square has a whole number square root.

You know how to find the square root of a perfect square.

What happens if the number is not a perfect square?

Does it have a square root?

What would the square root look like?

Think about the square root of 50.

Where would it be on this chart?

What can you say about the square root of 50?

50 is between the perfect squares 49 and 64 but closer to 49.

So the square root of 50 is between 7 and 8 but closer to 7.

Square Perfect Root Square 1 1 2 4 3 9 4 16 5 25 6 36 7 49 8 64 9 81 10 100 11 121 12 144 13 169 14 196 15 225

When estimating square roots of numbers, you need to determine:

Between which two perfect squares it lies (and therefore which 2 square roots).

Which perfect square it is closer to (and therefore which square root).

Example: 110

Lies between 100 & 121, closer to 100.

So 110 is between 10 & 11, closer to 10.

Square Perfect Root Square

1 12 43 94 165 256 367 498 649 8110 10011 12112 14413 16914 19615 225

Estimate the following:

30

200

215

Square Perfect Root Square

1 1 2 4 3 9 4 16 5 25 6 36 7 49 8 64 9 81 10 100 11 121 12 144 13 169 14 196 15 225

Approximating a Square Root

Approximate to the nearest integer

< <

< <6 7

Identify perfect squares closest to 38

Take square root

Answer: Because 38 is closer to 36 than to 49, is closer to 6 than to 7. So, to the nearest integer, = 6

Approximate to the nearest integer

Identify perfect squares closest to 70

Take square root

Identify nearest integer

< <

<<

Another way to think about it is to use a number line.

Since 8 is closer to 9 than to 4, √8 is closer to 3 than to 2, so √8 ≈ 2.8

2 2.2 2.4 2.6 2.8 3.0 2.1 2.3 2.5 2.7 2.9

√8

Example:

Approximate

10 10.2 10.4 10.6 10.8 11.0 10.1 10.3 10.5 10.7 10.9

A

B

D

C

9 and 16

36 and 49

25 and 36

49 and 64

41 The square root of 40 falls between which two perfect squares?

Identify perfect squares closest to 40

Take square root

Identify nearest integer

< <

<<

42 Which whole number is 40 closest to?

A

B

D

C

36 and 49

64 and 84

49 and 64

100 and 121

43 The square root of 110 falls between which two perfect squares?

44 Estimate to the nearest whole number.

110

45 Estimate to the nearest whole number.

219

46 Estimate to the nearest whole number.

90

47 What is the square root of 400?

48 Approximate to the nearest integer.

9649 Approximate to the nearest integer.

50 Approximate to the nearest integer.167

51 Approximate to the nearest integer.140

52 Approximate to the nearest integer. 40

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.

A

B

D

C

3 and 9

9 and 10

8 and 9

46 and 47

53 The expression is a number between

Rational & IrrationalNumbers

Return to Table of Contents

Rational & Irrational Numbers

is rational because the radicand (number under the radical) is a perfect square

If a radicand is not a perfect square, the root is said to be irrational.

Ex:

Sort the following numbers.

Rational Irrational

25 36 10064 200

625

24

1225

0

3600

4032 52 225

300 1681

A BRational Irrational

54 Rational or Irrational?

A BRational Irrational

55 Rational or Irrational?

A BRational Irrational

56 Rational or Irrational?

A BRational Irrational

57 Rational or Irrational?

A BRational Irrational

58 Rational or Irrational?

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.

A

B

D

C

p

59 Which is a rational number?

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.

60 Given the statement: “If x is a rational number, then is irrational.” Which value of x makes the statement false?

A

B

D

C 3

2

4

Radical Expressions Containing Variables

Return to Table of Contents

To take the square root of a variable rewrite its exponent as the square of a power.

Square Roots of Variables

=

=

= x12

(a8)

2

= a8

(x12)2

If the square root of a variable raised to an even power has a variable raised to an odd power for an answer, the answer must have absolute value signs. This ensures that the answer will be positive.

Square Roots of Variables

By Definition...

Examples

Try These.

= |x|5

= |x|13

no no

How many of these expressions will need an absolute value sign when simplified?

yes

yes

yes

yes

A

B

D

C

61 Simplify

A

B

D

C

62 Simplify

A

B

D

C

63 Simplify

A

B

D

C

64 Simplify

no real solution

A

B D

C

65

Simplifying Non-Perfect Square Radicands

Return to Table of Contents

What happens when the radicand is not a perfect square?

Rewrite the radicand as a product of its largest perfect square factor.

Simplify the square root of the perfect square.

When simplified form still contains a radical, it is said to be irrational.

Try These.

Identifying the largest perfect square factor when simplifying radicals will result in the least amount of work.

Ex:

Not simplified! Keep going!

Finding the largest perfect square factor results in less work:

Note that the answers are the same for both solution processes

already in simplified form

A

B

D

C

66 Simplify

already in simplified form

A

B

D

C

67 Simplify

already in simplified form

A

B

D

C

68 Simplify

69 Simplify

already in simplified form

A

B

D

C

70 Simplify

already in simplified form

A

B

D

C

71 Simplify

already in simplified form

A

B

D

C

Which of the following does not have an irrational simplified form?

72

A

B

D

C

Note - If a radical begins with a coefficient before the radicand is simplified, any perfect square that is simplified will be multiplied by the existing coefficient. (multiply the outside)

Express in simplest radical form.

73 Simplify

A

B

D

C

74

A

B

D

C

Simplify

75

A

B

D

C

Simplify

76

A

B

D

C

Simplify

77

A

B

D

C

Simplify

20

10

7

4

78

A

B

D

C

When is written in simplest radical form, the

result is . What is the value of k?

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.

79 When is expressed in simplestform, what is the value of a?

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.

6

2

3

8

A

B

D

C

Simplifying Roots of Variables

Return to Table of Contents

Simplifying Roots of Variables

Remember, when working with square roots, an absolute value sign is needed if:• the power of the given variable is even and• the answer contains a variable raised to an odd power

outside the radical

Examples of when absolute values are needed:

Simplifying Roots of Variables

Divide the exponent by 2. The number of times that 2 goes into the exponent becomes the power on the outside of the radical and the remainder is the power of the radicand.

Note:Absolute value signs are not needed because the radicand had an odd power to start.

Example

Only the y has an odd power on the outside of the radical.

The x had an odd power under the radical so no absolute value signs needed.

The m's starting power was odd, so it does not require absolute value signs.

Simplify

80

A

B

D

C

Simplify

A

B

D

C

81 Simplify

A

B

D

C

82 Simplify

A

B

D

C

83 Simplify

Properties of Exponents

Return to Table of Contents

Rules of Exponents

Exponential Table Questions.pdf

Exponential Table.pdf

Exponential Test Review.pdf

Materials

There are handouts that can be used along with this section. They are located under the heading labs on the Exponential page of PMI Algebra. Documents are linked. Click the name above of the document.

The Exponential Table

x 1 2 3 4 5 6 7 8 9 10

1

2

3

4

5

6

7

8

X X X X X X X X XX

Why is 24 equivalent to 4

2? Write the values out in

expanded form and see if you can explain why.

Question 1

x 1 2 3 4 5 6 7 8 9 10

1

2 16

3 72

9

4 16

5

6 72

9

7

8

X X X X X X X X XX

16 x 64 = 1024 42 x 4

3 = 4

5

Write the equivalent expressions in expanded form. Attempt to create a rule for multiplying exponents with the same base.

Question 2

x 1 2 3 4 5 6 7 8 9 10

1

2 16

3 64

4

5 102

4

6

7

8

X X X X X X X X XX

8 x 27 = 216 23 x 3

3 = 6

3

Write the equivalent expressions in expanded form. Attempt to create a rule for multiplying exponents with the same power.

Question 3

x 1 2 3 4 5 6 7 8 9 10

1

2

3 8 27 21

6

4

5

6

7

8

X X X X X X X X XX

15625 ÷ 625 = 25 56 ÷ 5

4 = 5

2

Write the equivalent expressions in expanded form. Attempt to create a rule for dividing exponents with the same base.

Question 4

x 1 2 3 4 5 6 7 8 9 10

1

2 25

3

4 625

5

6 1562

5

7

8

X X X X X X X X XX

A. 1. Explain why each of the following statements is true.

A. 23 x 2

2 = 2

5

(2 x 2 x 2) x (2 x 2) = (2 x 2 x 2 x 2 x 2)

B. 34 x 3

3 = 3

7

C. 63 x 6

5 = 6

8

am x a

n = a

m + n

B. 1. Explain why each of the following statements is true.

A. 23 x 3

3 = 6

3

(2 x 2 x 2) x (3 x 3 x 3) = (2 x 3)(2 x 3)(2 x 3)

B. 53 x 6

3 = 30

3

C. 104 x 4

4 = 40

4

am x b

m = (ab)

m

C. 1. Explain why each of the following statements is true.

A. 42 = (2

2)

2 = 2

4

B. 92 = (3

2)

2 = 3

4

C. 1252 = (5

3)

2 = 5

6

(am)

n = a

mn

D. 1. Explain why each of the following statements is true.

A. 35

32

46

B. 45

C. 510

510

=

=

=

33

41

50 a

m

an

= am-n

Operating with Exponents

32 x 3

4 = 3

6

52 x 3

2 = 15

2

am x a

n = a

m+n

am x b

m = (ab)

m

(43)

2 = 4

6

Examples

(am)

n = a

mn

am

= am-n

an

35

= 32

33

415

48

42

47

84

A

B

D

C

Simplify: 43 x 4

5

52

54

521

510

85

A

B

D

C

Simplify: 57 ÷ 5

3

86

A

B

D

C

Simplify: 47 x 5

7

87

A

B

D

C

Simplify:

88

A

B

D

C

Simplify:

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.

x2y

2z

3

x3y

3z

4

x4y

2z

5

89

A

B

D

C

The expression (x2z

3)(xy

2z) is equivalent to

90

A

B

D

C

Simplify:

simplified

91

A

B

D

C

Simplify:

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.

2w5

2w8

20w5

20w8

92

A

B

D

C

The expression is equivalent to

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.

-13

-23

-31

-85

93

A

B

D

C

If x = - 4 and y = 3, what is the value of x - 3y2?

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.

–3x2

3x2

–27x15

27x8

94

A

B

D

C

When -9 x5 is divided by -3x

3, x ≠ 0, the quotient is

By definition:

x-1 = , x 0

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.

x4

-4x

0

95

A

B

D

C

Which expression is equivalent to x-4?

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.

-6

-8

96

A

B

D

C

What is the value of 2-3?

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.

97

A

B D

C

Which expression is equivalent to x-1• y

2?

xy2

xy-2

Solving Equations with Perfect Square and Cube Roots

Return to Table of Contents

The product of two equal factors is the "square" of the number.

The product of three equal factors is the "cube" of the number.

When we solve equations, the solution sometimes requires finding a square or cube root of both sides of the equation.

When your equation simplifies to:

x2 = #

you must find the square root of both sides in order to find the value of x.

When your equation simplifies to:

x3 = #

you must find the cube root of both sides in order to find the value of x.

Example:

Solve.Divide each side by the coefficient. Then take the square root of each side.

Example:

Solve. Multiply each side by nine, then take the cube root of each side.

Try These:

Solve.

± 10 ±8 ±9 ±7

Try These:

Solve.

2 1 4 5

98 Solve.

99 Solve.

100 Solve.

101 Solve.