algebraic roots and radicals approximating square roots rational & irrational numbers radical...

Algebraic Roots and Radicals

Approximating Square Roots

Rational & Irrational Numbers

Radical Expressions Containing Variables

Simplifying Non-Perfect Square Radicands

Operations with Radicals

Simplifying Perfect Square Radical Expressions

Simplifying Roots of Variables

Pythagorean Theorem

Distance Formula

Intro to Trig

Solving Right Triangles

Click on topic to go to that

section.

Return to Table of Contents

Simplifying Perfect Square Radical Expressions

Can you recall the perfect squares from 1 to 169?

 12 =  82 =  

 22 = 92 =

 32 = 102 =

 42 =  112 =

 52 =  122 =

 62 = 132 = 202 =

 72 =

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?

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 expression

?

is not real

Evaluate the expression

1

2 ?

3 = ?

4

5

6 = ?

A 3

B -3

C No real roots

7 The expression equal to

is equivalent to a positive integer when b is

A -10

B 64

C 16

D 4

Square Roots of Fractions

ab = b 0

1649 = =

4

7

Try These

8

A

B

C

D no real solution

9

A

B

C

D no real solution

10

A

B

C

D no real solution

11

A

B

C

D no real solution

12

A

B

C

D no real solution

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

13 Evaluate

A B

C D No Real Solution

14 Evaluate

A .06 B .6

C 6 D No Real Solution

15 Evaluate

A .11 B 11

C 1.1 D No Real Solution

16 Evaluate

A .8 B .08

C D No Real Solution

17 Evaluate

A B

C D No Real Solution

ApproximatingSquare Roots

Return to Table of Contents

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

< <

<<

18 Approximate to the nearest integer

19 Approximate to the nearest integer

20 Approximate to the nearest integer

21 Approximate to the nearest integer

22 Approximate to the nearest integer

23 The expression is a number between

A 3 and 9

B 8 and 9

C 9 and 10

D 46 and 47

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 by the square root being rational or irrational.

24 Rational or Irrational?

A Rational B Irrational

25 Rational or Irrational?

A Rational B Irrational

26 Rational or Irrational?

A Rational B Irrational

27 Rational or Irrational?

A Rational B Irrational

28 Rational or Irrational?

A Rational B Irrational

29 Which is a rational number?

A

B p

C

D

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

A

B 2

C 3

D 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

=

=

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.

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

31 Simplify

A

B

C

D

32 Simplify

A

B

C

D

33 Simplify

A

B

C

D

34 Simplify

A

B

C

D

35

A

B

C

D no real solution

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

36 Simplify

A

B

C

D already in simplified form

37 Simplify

A

B

C

D already in simplified form

38 Simplify

A

B

C

D already in simplified form

39 Simplify

A

B

C

D already in simplified form

40 Simplify

A

B

C

D already in simplified form

41 Simplify

A

B

C

D already in simplified form

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

A

B

C

D

2

43 Simplify

A

B

C

D

44 Simplify

A

B

C

D

45 Simplify

A

B

C

D

46 Simplify

A

B

C

D

47 Simplify

A

B

C

D

48 When is written in simplest radical form, the result is .What is the value of k?

A 20

B 10

C 7

D 4

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

A 6

B 2

C 3

D 8

Express −3 48 in simplest radical form.

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 andthe 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

50 Simplify

A

B

C

D

Pull

51 Simplify

A

B

C

D

52 Simplify

A

B

C

D

53 Simplify

A

B

C

D

Operations with Radicals

Return to Table of Contents

Adding or Subtracting Radicals

Radicals can be added and subtracted when they have like terms.

Like Terms means they have the same radicands.

Like Terms Unlike Terms

54 Identify the like terms

A

B

C

D

E

F

To add or subtract radicals, add or subtract the coefficients; the radicand remains the same.

Examples.

Try These.

55 Simplify

A

B

C

D Already Simplified

56 Simplify

A

B

C

D Already Simplified

57 Simplify

A

B

C

D Already Simplified

58 Simplify

A

B

C

D Already Simplified

59 Simplify

A

B

C

D Already Simplified

Radicals must be simplified before adding or subtracting

60 Simplify

A

B

C

D Already in simplest form

61 Simplify

A

B

C

D Already in simplest form

62 What is the sum of and   ?

A

B 7

C 9

D 29

63 What is the sum of and ?

A

B

C

D

64 The expression -is equivalent to

A

B 10

C

D

65 Simplify

A

B

C

D Already in simplest form

66 Which of the following expressions does not equal the other 3 expressions?

A

B

C

D

Multiplying Radicals

To multiply radicals, multiply the coefficients then multiply the radicands. Simplify if possible.

Multiplying Radicals

coefficient times coefficient and radicand times radicand

67 Multiply

A

B

C

D

Multiplying Radicals

After multiplying, check to see if radicand can be simplified.

68 Simplify

A

B

C

D

69 Simplify

A

B

C

D

70 Simplify

A

B

C

D

71 Simplify

A

B

C

D

Multiplying Polynomials Involving Radicals  1) Follow the rules for distribution. 2) Be sure to simplify radicals when possible and combine like terms.

72 Multiply and write in simplest form:

A

B

C

D

73 Multiply and write in simplest form:

A

B

C

D

74 Multiply and write in simplest form:

A

B

C

D

75 Multiply and write in simplest form:

A

B

C

D

76 Multiply and write in simplest form:

A

B

C

D

Rationalizing the Denominator

Which of these expressions has a rational denominator?

RationalDenominator

IrrationalDenominator

A simplified fraction does not have a radical in the denominator.

The process of eliminating a radical in the denominator is called "rationalizing the denominator".

To rationalize the denominator, you create an equivalent fraction by multiplying the numerator & denominator by the denominator's radical.

Examples.

77 Simplify

A

B

C

D Already simplified

78 Simplify

A

B

C

D Already simplified

79 Simplify

A

B

C

D Already simplified

80 Simplify

A

B

C

D Already simplified

81 Simplify

A

B

C

D Already simplified

Pythagorean Theorem

Return to the Table of Contents

Recall...

right triangleis a triangle with a right angle.

The sides form that right angle are the legs.The side opposite the right angle is the hypotenuse.

The hypotenuse is also the longest side.

leg

hypotenuseleg

Pythagorean Theorem (R1)In a right triangle, the sum of the squares of the lengths of the legs

is equal to the square of the length of the hypotenuse.

leg2 + leg2 = hypotenuse2ora2 + b2 = c2 a

b

c

Example

Find the length of the missing side of the right triangle.

x9

12

Is the missing side a leg or the hypotenuse of the right triangle?

hypotenuse

92 + 122 = x281 + 144 = x2225 = x215 = x

-15 is a extraneous solution, a distance can not equal a negative number.

x = 15

x9

12

Example

Find the length of the missing side.

Is the missing side a leg or the hypotenuse of the right triangle?

leg

x

28 20

x2 + 202 = 282x2 + 400 = 784x2 = 384x = 8 6

x

28 20

82 The missing side is the ________ of the right triangle.

A leg

B hypotenuse6 9

x

83 Find the length of the missing side.

6 9

x

84 The missing side is the _________ of the right triangle.

A leg

B hypotenusex

15

36

85 Find the length of the missing side.

x15

36

The safe distance of the base of the ladder from a wall it leans against should be one-fourth of the length of the ladder.

28 feet

7 feet

?

Thus, the bottom of a 28-foot ladder should be 7 feet from the wall. How far up the wall will a the ladder reach?

28 feet

7 feet

?

a2 + b2 = c272 + b2 = 28249 + b2 = 384b2 = 335b 18.30

The ladder will reach 18.3 feet up the wall safely.

84

50 x

Try this...

The dimensions of a high school basketball court are 84' long and 50' wide. What is the length of from one corner of the court to the opposite corner?

842 + 502 = x29556 = x297.75 = x

The court is 97.75 feet

An

sw

er

86 A NBA court is 50 feet wide and the length from one corner of the court to the opposite corner is 106.5 feet. How long is the court?

A 94.03 feet

B 117.7 feet

C 118 feet

D 94 feet

(Round the answer to the nearest whole number)

Pythagorean Theorem Applications

The Pythagorean Theorem can also be used in figures that contain right angles.

ExampleFind the perimeter of the square.

Before finding the perimeter of the square, we need to first find the length of each side.

18 cmPsq = 4s

18 cmx

Remember, in a square all sides are congruent.

x2 + x2 = 1822x2 = 324x2 = 162x2 = 9 2

P = 4sP = 4(9 2)P = 36 2 cm

ExampleFind the area of the triangle.

The base of the triangle is given, but we need to find the height of the triangle.

A = bh12

13 feet

10 feet

13 feet

By definition, the altitude (or height) of an isosceles triangle is the perpendicular bisector of the base.

13 feet

5 feet

13 feeth

5 feet

52 + h2 = 13225 + h2 = 169h2 = 144h = 12

A = (10)(12)A = (120)A = 60 feet 1

212

AN

SW

ER

Try this...Find the perimeter of the rectangle.

Prect = 2l + 2w

8 in

10 in

87 Find the area of the rectangle.

A 120 feet

B 84 feet

C 46 inches

D 46 feet

8 feet17 fe

et

88 Find the perimeter of the square. (Round to the nearest tenth)

A 25.46 cm

B 25.4 cm

C 25.5 cm

D 25.6 cm

9 cm

89 Find the area of the triangle.

7 inches

24 inches

7 inches

Converse of the Pythagorean Theorem (R2)If the square of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the

triangle is a right triangle.

If c2 = a2 + b2, then ABC is a right triangle. a

b

c

A

B

C

ExampleTell whether the triangle is a right triangle.

c2 = a2 + b2252 = 72 + 242625 = 49 + 576 625 = 625

 DEF is a right triangle.

Remember c is the longest side

D E

F

7

24

25

Theorem (R3)If the square of the longest side of a triangle is greater than

the sum of the squares of the other two sides, then the triangle is obtuse.

If c2 > a2 + b2, then ABC is obtuse.

A

B

C

a

b

c

Theorem (R4)If the square of the longest side of a triangle is less than the

sum of the squares of the other two sides, then the triangle is acute.

If c2 < a2 + b2, then ABC is acute.

a

b

c

A

B

C

ExampleClassify the triangle as acute, right, or obtuse.

c2 ? a2 + b2172 ? 152 + 132289 ? 225 + 169289 < 394

The triangle is acute.

17

15 13

ExampleTell whether 12, 3, 3 15 represent the sides of a acute, right, or obtuse triangle.

First, we need to find the approximate value of 3 15, to determine if 3 15 or 12 is the longest side.

3 15 11.62, so 12 is the longest side.

122 ? 32 + (3 15)2144 ? 9 + 135144 = 144

The triangle is right.

90 Classify the triangle is acute, right, obtuse, or not a triangle.

A acute

B right

C obtuse

D not a triangle

11

12

15

91 Classify the triangle is acute, right, obtuse, or not a triangle.

A acute

B right

C obtuse

D not a triangle

6

3

5

92 Classify the triangle is acute, right, obtuse, or not a triangle.

A acute

B right

C obtuse

D not a triangle

25

19

20

93 Tell whether the lengths 35, 65, and 56 represent the sides of an acute, right, or obtuse triangle.

A acute

B right

C obtuse

94 Tell whether the lengths represent the sides of an acute, right, or obtuse triangle.

A acute triangle

B right triangle

C obtuse triangle

Review

If c2 = a2 + b2, then triangle is right.

If c2 < a2 + b2, then triangle is acute.

If c2 > a2 + b2, then triangle is obtuse.

Distance

Return to Table of Contents

Computing the distance between two points in the plane is an application of the Pythagorean Theorem for right triangles.

Computing distances between points in the plane is equivalent to finding the length of the hypotenuse of a right triangle.

(x1, y1) (x2, y1)

(x2, y2)

The distance formulacalculates the distance using point's coordinates.

c

Relationship between the Pythagorean Theorem & Distance Formula

c

b

a

The Pythagorean Theorem states a relationship among the sides of a right triangle.

c2= a2 + b2

The Pythagorean Theorem is true for all right triangles. If we know the lengths of two sides of a right triangle then we know the length of the third side.

Distance

The distance between two points, whether on a line or in a coordinate plane, is computed using the distance formula.

The Distance Formula

The distance 'd' between any two points with coordinates and is given by the formula:(x1, y1) (x2, y2)

d =

Note: recall that all coordinates are (x-coordinate, y-coordinate).

Example

Calculate the distance from Point K to Point I

(x1, y1) (x2, y2)

d =

Plug the coordinates into the distance formula

Label the points - it does not matterwhich one you label point 1 and point 2. Your answer will be the same.

KI =

KI = =

=

95 Calculate the distance from Point J to Point K

A

B

C

D

96 Calculate the distance from H to K

A

B

C

D

97 Calculate the distance from Point G to Point K

A

B

C

D

98 Calculate the distance from Point I to Point H

A

B

C

D

99 Calculate the distance from Point G to Point H

A

B

C

D

Trigonometric Ratios

Return to theTable of Contents

Trigonometry - is a branch of mathematics that deals with relationship of the sides and angles of

triangles.

A trigonometric ratio is the ratio of the two lengths of a right triangle.

There are 3 ratios for each acute angle of a right triangle.

The ratios are called sine, cosine, and tangent abbreviated sin, cos, and tan respectively.

a

b c

A

BC

a

b c

A

BC

sinθ = side opposite hypotenuse

cosθ =side adjacent hypotenuse

tanθ = side oppositeside adjacent

SOHCAH

TOA

SOHCAHTOA

a

b c

A

BC

In each right triangle, there are 2 acute angles. In the triangle to the left <A and <B are the acute angles.

Let's look at <A.Find the side opposite, side adjacent, hypotenuse.

a

b c

A

BC

The side opposite <A is a.The hypotenuse is c.The side adjacent (or next to) <A is b.

sinA = side opposite <A hypotenuse

ac=

cosA = side adjacent <A hypotenuse

bc=

tanA =side opposite <Aside adjacent to <A

ab=

SOHCAHTOA

Example

Find the sin, cos, and tan of <F.

D

E F

6

8

10

What is the side opposite, side adjacent, and the hypotenuse of the right triangle?

D

E F

6

8

10

DF is the hypotenuse. DE is the side opposite to < F.EF is the side adjacent to <F.

sinF =opphyp

6 10=

3 5= cosF =

adjhyp

8 10=

4 5

=

tan F = oppadj

6 8= 3

4=

100 What is the side opposite to <J?

A JL

B LK

C KJ

J

K

L

101 What is the hypotenuse of the triangle?

A JL

B LK

C KJ

J

K

L

102 What is the side adjacent to <J?

A JL

B LK

C KJ

J

K

L

103 What is the sinR?

A 9/13

B 7/9

C 7/13

D 9/7

Q

R

S

13

7

9

104 What is the cosR?

A 9/13

B 7/9

C 7/13

D 9/7

Q

R

S

13

7

9

105 What is the tanR?

A 9/13

B 7/9

C 7/13

D 9/7

Q

R

S

13

7

9

Using Trigonometric Ratios to find side length.

12

G E

M

25o

65o

x

When solving right triangles, you can use either acute angle to find the answer.

(You will need a calculator or trig table)

In the triangle, the length of GM is given and EM is the side we

need to find.12

G E

M

25o

65o

x

Referring to <G.EM is the side opposite and GM is the hypotenuse.

The trig ratio that trig ratio uses the side opposite and hypotenuse, is the sine function.

sin G =

sin25 =

.4226 =

x ≈ 5.07

EMGM

x 12

x 12(12) (12)

12

G E

M

25o

65o

x

12

G E

M

25o

65o

x

Referring to <M.EM is the side adjacent and GM is the hypotenuse.

The trig ratio that uses the side adajacent and the hypotenuse, is the cosine function.

12

G E

M

25o

65o

x

cos M =

cos 65 =

x ≈ 5.07

.4226 =

EMGM

x 12

x 12

(12)(12)

C

AE y

10

70o

20o

Referring to <C.EA is the side opposite and CE is the side adjacent.

Referring to <A.CE is the side opposite and EA is the side adjacent.

The trig ratio that uses the side opposite and the side adjacent, is the tangent function.

In the triangle, the length of CE is given and EA is the side we

need to find.

C

AE y

10

70o

20o

tanC =

tan70 =

2.747 =

y ≈ 27.47

EACE

y 10

(10) y 10(10)

tan A =

tan 20 =

.3640 =

.3640y = 10

y ≈ 27.47

CEEA

10 y

(y)10 y(y)

106 Evaluate sin60. Round to the nearest ten-thousandth.

107 Evaluate cos45. Round to the nearest ten-thousandth.

108 Evaluate tan30. Round to the nearest ten-thousandth.

109 Using <B, which is the correct ratio needed to solve for x.

A sin40 = 12/x

B cos40 = x/12

C tan40 = 12/x

D sin40 = x/12

B

E

x12

40o

50oD

110 Using <D, which is the correct ratio needed to solve for x.

A sin50 = 12/x

B cos50 = x/12

C tan50 = 12/x

D sin50 = x/12

B

E

x12

40o

50oD

111 Using <J, which is the correct ratio needed to solve for y.

A tan32 = x/11

B cos32 = x/11

C tan32 = 11/x

D sin32 = 11/x

J L

K

x

32o

58o11

112 Using <K, which is the correct ratio to solve for y.

A tan58 = x/11

B cos58 = x/11

C tan58 = 11/x

D sin 58 = 11/xJ L

K

x

32o

58o11

113 Find the length of LM.

LM

P

12

68o22o

114 Find the length of LP.

LM

P

12

68o22o

Solving Right Triangles

Return to the Table of Contents

To solve a right triangle means to find the length of each side and the measure of each angle in the triangle.

When using trigonometric ratios to solve a right triangle, you need to know either the length of 2 sides or the length of one side and the

measure of one the acute angles.

Rememberm<A + m<B + m<C = 180o

a2 + b2 = c2SOHCAHTOA

In this section you will need to use the inverse trig function to solve the equations. Just as the following are inverses and undo

each other,Addition Subtraction

Multiplication DivisionSquare Square Root

so does a trig ratio and its inverse.sinθ sin-1θcosθ cos-1θtanθ tan-1θ 

inverse

inverse

inverse

inverse

inverse

inverse

115 Find sin-10.8. Round to the nearest hundredth.

116 Find tan-12.3. Round to the nearest hundredth.

117 Find cos-10.45. Round to the nearest hundredth.

In ABC we need to find the m<A, m<C and BC.

Referring to <C, AB is the side opposite and AC is the hypotenuse

Referring to <A, AB is the side adjacent and AC is the hypotenuse

Which functions should be used to find the m<C and m<A?

9

15

A B

C

CH

EC

K

m<A + m<B + m<C = 180o53.13o + 90o + 36.87o = 180o

180o = 180o

9

15

A B

C

To find the m<C, use the sin function.

sinC =

sinC =

sinC = 0.6

sin-1C ≈ 36.87

m<C ≈36.87o

ABAC

9 15

To find the m<A, usethe cos function.

cosA =

cosA =

cos A = 0.6

cos-1A ≈53.13

m<A ≈ 53.13o

ABAC

9 15

9

15

A B

C

Since two sides of the triangle is given, to find BC use the Pythagorean Theorem.

a2 + b2 = c292 + x2 = 15281 + x2 = 225

x2 = 144x = 12

BC = 12

Try this...Solve the right triangle. Round your answers to the nearest hundredth.

Q

R

S

724

QS = 25m<Q = 73.74om<R = 16.26o

Click to Reveal Answer

118 Find CE.

C

D E

8

5

119 Find m<C.

C

D E

8

5

120 Find the m<E.

C

D E

8

5

Find the missing parts of the triangle.

L

A

B

64o

15

Referring to <L, AB is the side opposite and AL is the hypotenuse.

Which trig function must be used?

sin L =

sin64 =

.8988 =

z ≈ 13.48

ABAL

z 15

z 15

AB ≈ 13.48

L

A

B

64o

15

m<L + m<A = 90o64o + m<A = 90om<A = 26o

a2 + b2 = c2a2 + (13.48)2 = 152a2 + 181.79 = 225a2 = 43.29a ≈ 6.58

L

A

B

z

64o

15

Try this...Find the missing parts of the triangle.

R

E D

37o

RD ≈ 18.09ED ≈ 14.36m<R = 53o

Click to Reveal Answer

121 Find the m<G.

L

A G18

20o

122 Find AL.L

A G18

20o

123 Find the m<P.

A 49.19o

B 33.69o

C 41.81o

D 56.31o

P

EN

12

18

124 Find RT.

A 10.44

B 12.45

C 11.47

D 9.53

40o

S

R

T

8