Ch. 4 Notes - Mr. McDonald'smrmcdonaldshomepage.weebly.com/.../1/3/9/2/13921997/ch._4__radicals_.pdf · Math 2201 Unit 4: Radicals Read Learning Goals, p. 173 text. Ch. 4 Notes 4.1

Math 2201

Unit 4: Radicals

Read Learning Goals, p. 173 text.

Ch. 4 Notes

§4.1 Mixed and Entire Radicals (1 class)

Read Goal p. 176 text.

Outcomes:

1. Define and give an example of a radical. See notes

2. Identify the index and the radicand of a radical. See notes

3. Define and give an example of an entire radical. pp. 175, 515

4. Define and give an example of a mixed radical. pp. 175, 516

5. Define and give an example of a perfect square. pp. 175, 517

6. Define and give an example of a principal square root. pp. 176, 517

7. Define and give an example of a secondary square root. pp. 176, 517

8. Simplify a simple radical. p. 178.

9. Convert an entire radical to a mixed radical. p. 178

10. Convert a mixed radical to an entire radical. p. 179

nDef : A radical is any expression that can be written in the form n b where:

n is called the index of the radical, and

b is called the radicand.

E.g.: 24 4 (index is 2, radicand is 4) E.g.: 27 7 (index is 2, radicand is 7)

E.g.: 3 16 (index is 3, radicand is 16) E.g.: 4 100 (index is 4, radicand is 100)

E.g.: 532

35

32index is 5, radicand is

35

E.g.: 4x (index is 2, radicand is 4x )

E.g.: 127

1

2x 121

index is 7, radicand is 2

x

nDef : A perfect square is a whole number that is the square of another whole number.

E.g.: If we square 6, we get 36. So 36 is a perfect square.

E.g.: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225 are perfect squares.

Finding the Square Root of a Number

Recall that numbers usually have two square roots, one positive and one negative.

E.g.: The square roots of 16 are 4 and -4 because 2 2

4 16 and 4 16 .

nDef : The principal square root of a number is the positive square root.

E.g.: 4 is the principal square root of 16 because 16 4 .

nDef : The secondary square root of a number is the negative square root.

E.g.: -4 is the principal square root of 16 because 16 4 .

Note that the principal square root is most commonly used in the real world.

Do # 1 (omit c), p. 182 text in your homework booklet.

Simplifying Simple Radicals

E.g.: Simplify 144

144 12

Why isn’t 144 12 ?

E.g.: Simplify 3 27

3 27 3

E.g.: Simplify 4 38416

4 38416 14

Why isn’t 4 38416 14


5 32768 8

nDef : Entire radicals are radicals with a coefficient of 1.

E.g.: 56 1 56 is an entire radical.

E.g.: 3 3108 1 108 is an entire radical.

E.g.: 2 24 41 1

8 1 82 2

x x

is an entire radical.

nDef : Mixed radicals are radicals where the coefficient is NOT 1. They are written in the form nk b

where k is NOT 1.

E.g.: 2 5 is a mixed radical because the coefficient is 2.

E.g.: 5 7 is a mixed radical because the coefficient is -5.

E.g.: 5

47

a is a mixed radical because the coefficient is 5

7 .

Converting Mixed Radicals to Entire Radicals

You may be asked to convert a mixed radical into an entire radical. To do this, you should make use of

inverse operations like squaring and square root, cubing and cube root, and so on. For example,

3 4 52 3 543 547 7 49, 6 6 216, 11 11 14641, 5 5 3125

E.g.: Convert 5 2 to an entire radical.

25 2 5 2 25 2 25 2 50


3 33 3 33 3 34 2 4 2 64 2 64 2 128


444 4 4 4 4 43 6 3 6 81 6 81 6 486


55 5 55 5 558 2 8 2 32768 2 32768 2 65536

Comparing and Ordering Radicals

To help order and compare radicals, it is often useful to write all radicals as entire radicals and compare

the radicands.

E.g.: Order from least to greatest: 1

26 7 , 7 2, 20, 20 3, 401

First convert each mixed radical to an entire radical.

1

26 7 6 7 36 7 36 7 252

7 2 49 2 49 2 98

20 400

20 3 400 3 400 3 1200

401

The entire radicals from least to greatest are 98, 252, 400, 401, 1200 , so the original radicals

from least to greatest are 1

27 2, 6 7 , 20, 401, 20 3

Note that you could have also changed each radical to a decimal using a calculator and used the decimal

approximations to order the radicals.

Do # 11, p. 183 text in your homework booklet.

Converting an Entire Radical to a Mixed Radical

To convert entire radicals to mixed radicals, you often use perfect squares like 4, 9, 16, 25, 36, 49, 64,

81, 100, 121, 144, …, perfect cubes like 8, 27, 64, 125, …, and so on.

E.g.: Convert 128 to a mixed radical.

Find the largest perfect square that divides evenly into 128.

128 64 2 8 2

E.g.: Convert 3 250 to a mixed radical.

Find the largest perfect cube that divides evenly into 250.

3 33 33250 125 2 125 2 5 2

Do #’s 2, 3 a, 4 a, c, 5, p. 182 text in your homework booklet.

§4.2 Adding and Subtracting Radicals (2 classes)


Outcomes:

1. Define and give an example of like radicals. See notes

2. Add and subtract like radicals. pp. 184-187

3. Simplify radical expressions. pp. 184-187

nDef : Like radicals are radicals with the same index and the same radicand.

E.g.: 2 4 and 5 4 are like radicals because both have an index of 2 and a radicand of 4.

E.g.: 3 32 7 and 3 7 are like radicals because both have an index of 3 and a radicand of 7.



E.g.: 32 4 and 5 4 are NOT like radicals. Why not?

E.g.: 8 85 5

1 7 3 and 3

2 4 4x x are NOT like radicals. Why not?

E.g.: 7 811 11

1 3 3 and 3

2 4 4x x are NOT like radicals. Why not?

Do # 1, p. 188 text in your homework booklet.

Adding and Subtracting Radicals

Adding and subtracting radicals is the same as adding and subtracting monomials. Only like radicals can

be added or subtracted, just as only like terms can be added or subtracted with monomials.

E.g.: Simplify 2 5 8x x x

Following the order of operations gives 2 5 8 7 8 1x x x x x x x

E.g.: Simplify 2 6 5 6 8 6

Following the order of operations gives 2 6 5 6 8 6 7 6 8 6 1 6 6

You can check your answer using a calculator.

Sometimes you may want to first simplify some or all of the radicals and then combine like radicals.

E.g.: Simplify 3 8 5 72 8 18

Simplifying the radicals and combining like radicals gives

3 4 2 5 36 2 8 9 2

3 2 2 5 6 2 8 3 2

6 2 30 2 24 2

36 2 24 2

12 2


E.g.: Simplify 75 4 7 252 48


25 3 4 7 36 7 16 3

5 3 4 7 6 7 4 3

9 3 2 7


E.g.: Simplify 3 3 3 31 2 3 432 135 500 1715

2 3 5 7


3 33 3 3 3 3 3

3 33 3

3 33 3

3 3

1 2 3 48 4 27 5 125 4 343 5

2 3 5 7

1 2 3 42 4 3 5 5 4 7 5

2 3 5 7

4 2 5 3 4 4 5

2 4 2 5


Do #’s 2 b, d, 4, 5 b, d, 6 b, d, 9 a, 11, 14-16, 18, 19, pp. 188-190 text in your homework booklet.

§4.3 Multiplying and Dividing Radicals (3 classes)


Outcomes:

1. Multiply radicals. pp. 192-196

2. Divide radicals. pp. 192-196

3. Explain what is meant by rationalizing the denominator. pp. 195, 517

4. Simplify radical expressions. pp. 192-196

Use a calculator to complete the following table.

n na b n a b

5 3 5 3 15

11 4 11 4 44

3 34 8 3 34 8 32

4 42 10 4 42 10 20

5 56 7 55 6 7 42

Make a conjecture about the value of n na b and the value of n a b .

Conjecture: They are ________________________.

Multiplying Radical Expressions

When multiplying radical expressions, we multiply the coefficients and then multiply the radicands. To

multiply radicals, they must have the same index.

E.g.: Simplify 2 8 6 108

Since each radical has an index of 2, we have the choice of multiplying the radicals as they appear and

then simplifying or simplifying each radical first and then multiplying. We’ll do this example both ways.

Method 1: Multiply first, simplify next.

2 8 6 108

2 6 8 108

12 864

12 144 6

12 144 6

12 12 6 144 6

Method 2: Simplify first, multiply next.

2 8 6 108

2 4 2 6 36 3

2 2 2 6 6 3

4 2 36 3

4 36 2 3 144 6

Do #’s 1, 4, p. 198 text in your homework booklet.

E.g.: Simplify 6 5 7 3 8 5

Since each radical has an index of 2, we can multiply these radicals using the distributive property.

6 5 7 3 8 5

6 5 7 3 6 5 8 5

6 7 5 3 6 8 5 5

42 15 48 25

42 15 48 5

42 15 240

Your Turn

Simplify 2 3 7 27 5 ANS: 126 2 15

Do #’s 5 a-d, p. 198 text in your homework booklet.

E.g.: Simplify 5 3 8 9 7 7 21

Since each radical has an index of 2, we can multiply these radicals using FOIL, or the box method, or

the happy rainbow method, etc.

5 3 8 9 7 7 21

5 3 9 7 5 3 7 21 8 9 7 8 7 21

45 21 35 63 72 7 56 21

11 21 35 9 7 72 7

11 21 35 3 7 72 7

11 21 105 7 72 7

11 21 33 7

E.g.: Simplify 3 8 4 2 7 3



3 8 4 2 7 3

3 8 2 3 8 7 3 4 2 4 7 3

6 8 21 24 8 28 3

6 4 2 21 4 6 8 28 3

6 2 2 21 2 6 8 28 3

12 2 42 6 8 28 3

E.g.: Simplify 20 24 3 12 5 32

Sometimes it may be better to simplify the radicals before multiplying. Since each radical has an index

of 2, we can multiply these radicals using FOIL, or the box method, or the happy rainbow method, etc.

Sometimes it may be better to simplify the radicals before multiplying.

20 24 3 12 5 32

4 5 4 6 3 4 3 5 16 2

2 5 2 6 3 2 3 5 4 2

2 5 2 6 6 3 20 2

2 5 6 3 2 5 20 2 2 6 6 3 2 6 20 2

12 15 40 10 12 18 40 12

12 15 40 10 12 9 2 40 4 3

12 15 40 10 12 3 2 40 2 3

12 15 40 10 36 2 80 3

Do #’s 5 e, p. 198 text in your homework booklet.

Sample Exam Question

Express 2

3 2 in simplest form.

a) 1

b) 5 2 6

c) 1 2 3

d) 5

E.g.: Simplify 2

5 3 8 2



5 3 8 2 5 3 8 2

5 3 5 3 5 3 8 2 8 2 5 3 8 2 8 2

25 9 40 6 40 6 64 4

25 9 80 6 64 4

25 3 80 6 64 2

75 80 6 128

203 80 6

Your Turn

Simplify 2

2 6 ANS:8 4 3

Do #’s 5 f, 11, p. 198 text in your homework booklet.

Use a calculator to complete the following table.

n

n

a

b n

a

b

152.2361...

3

155 2.2361...

3

221.4142...

11

222 1.4142...

11

3

3

8

4 33

82

4

4

4

10

2 44

105

2

5

5

642

2 55

6432 2

2

Make a conjecture about the value of n

n

a

b and the value of n

a

b.

Conjecture: They are ________________________.

Dividing Radical Expressions

When dividing radical expressions, we divide the coefficients and then divide the radicands. To divide

radicals, they must have the same index.

E.g.: Simplify 8 25

2 4

Since each radical has an index of 2, we can divide these radicals.

8 25 8 25 8 25 8 5 4010

2 2 4 2 2 42 4 4


16 7


64 56 64 56 564 4 8 4 4 2 4 2 2 8 2

16 716 7 7

E.g.: Simplify 3

3

108 18

12 9


3 3

333 3

108 18 108 18 189 9 2

12 912 9 9

E.g.: Simplify 4

4

98 56

49 8


4 444

4 4

98 56 98 56 562 2 7

49 849 8 8

E.g.: Simplify 5

5

11 3072

55 3


5 5

555 5

11 3072 11 3072 1 3072 1 1 41024 4

55 5 3 5 5 555 3 3

Rationalizing the Denominator with One Term in the Denominator

By convention, when we simplify an expression containing radicals, we do not leave a radical in the

denominator. Instead, we change the irrational number to a rational number. This is called rationalizing

the denominator.

E.g.: Simplify 4

9 3

We want an equivalent fraction that does NOT have the radical in the denominator. To do this we will

multiply both the numerator and the denominator by 3 .

4 3 4 3 4 3 4 3

9 3 279 3 3 9 9

You can check by changing both 4

9 3 and

4 3

27 to a decimal approximation.

E.g.: Simplify 2

5

We have to change the irrational number in the denominator into a rational number.

2 2 5 2 5 2 5 2 5

55 5 5 255 5

We changed the irrational denominator 2 1.414213562 into a rational number (5).

E.g.: Simplify 5 2

7

5 2 5 2 7 5 2 7 5 14 5 14

77 7 7 7 7 49

E.g.: Simplify 2

8

2 2 22 2 8 2 8 2 4 2 4 2 1 2 2

8 8 2 28 8 8 8 8 64


8 8

36 1236 18 36 18 8 36 18 8 36 18 8 36 144

8 88 8 8 8 8 8 8 8 8 8 8 8 64

432 216 108 54 27

64 32 16 8 4

Do #’s 2, 13, 14, 16 a, 19 a, 21 b, pp. 198-200 text in your homework booklet.

E.g.: Simplify 3 2 5

2

3 2 5 23 2 5 3 2 5 2 3 2 2 5 2 3 4 5 2 6 5 2

2 22 2 2 2 2 4

Alternative Solution

We could have broken the expression into two fractions first and then simplified.

3 2 5 3 2 5 3 2 2 5 2 3 2 2 5 2 3 4 5 2

2 2 2 2 2 2 2 2 2 2 2 4 4

3 2 5 2 6 5 2 6 5 2

2 2 2 2 2

E.g.: Simplify 6 2 3

3

6 2 3 6 2 3 3 6 2 3 3 3 6 6 9 6 6 3

33 3 3 3 3 9

Alternative Solution

6 2 3 6 2 3 6 2 3 6 2 3 6 6 6 6 6 6 31 1 1 1

3 3 33 3 3 3 3 3 3 9

6 6 3

3

Do # 16 b, c, d, p. 199 text in your homework booklet.

Do #’s 2, b, f, 3 b, d, f, 4, 6 b, d, e, 7, b, e, 8 b, c, e, 10 a, c, e, 11, 12 a, c, d, p. 203 text in your home

work booklet.

§4.4 Simplifying Algebraic Expressions Involving Radicals (2 classes)


Outcomes:

1. Simplify radical expressions containing variables. pp. 206-210

2. Identify the restrictions on the variable(s) in a radical expression. pp. 204, 517

Up to now, all our expressions involving radicals contained only numbers. Now we have to apply what

we have learned to radical expressions with variables.

nDef : The restrictions on an expression are the values for which the expression is defined.

E.g.: The restriction on 1

2x is 2x because

1

2x is defined for any real number except for 2.

E.g.: The restriction on x is 0x because x is NOT defined for numbers less than 0.

E.g.: The restriction on 6x is 6x because 6x is NOT defined for numbers less than 6.

E.g.: The restriction on 2x is 2x because 2x is NOT defined for numbers less than -2.

E.g.: There are no restrictions on 2x because 2x is defined for all real numbers.

E.g.: There are no restrictions on 4x because 4x is defined for all real numbers.

E.g.: There are no restrictions on 3 x because 3 x is defined for all real numbers.



Do #’s 1, 11, pp. 211, 213 text in your home work booklet.

Simplifying Radicals with Variables

Keep in mind that when variables are involved you must indicate the restrictions on the variable. To

determine the restrictions you must look at the original expression.

E.g.: Simplify 36 24x

3 3 26 24 6 24 6 4 6 6 2 6 12 6 ; 0x x x x x x x x x

E.g.: Simplify 3 58 96z z

3 5 3 5 3 4 3 2 3 2

5 5

8 96 8 96 8 16 6 8 4 6 8 4 6

32 6 32 6 ; 0

z z z z z z z z z z z z z

z z z z z

E.g.: Simplify 4 32y

4 32 4 8 4 8 2 8; 8y y y y y

Do # 2, p. 211 text in your home work booklet.

Adding and Subtracting Radical Expressions with Variables

Adding and subtracting radicals with variables is the same as adding and subtracting radicals without

variables. Only like radicals can be added or subtracted, just as only like terms can be added or

subtracted with monomials.

E.g.: Simplify 3 5x x

Since 3 and 5x x are like radicals (same index, same radicand) they can be combined.

3 5 8 ; 0x x x x

E.g.: Simplify 4 44 16 32x x

4 4 4 4 2 2 2 24 16 32 4 16 16 2 4 4 4 2 16 4 2 ;x x x x x x x x x

E.g.: Fill in the missing steps in the simplification below.

39 4

Step 1: 9 4

Step 2: 9 4

Step 3:

Step 4:

z z z

z z z

z z z

z

z

Do #’s 3 a, b, 5, 6a, 9 b, p. 212 text in your home work booklet.

Multiplying Radical Expressions with Variables

E.g.: Simplify 27 8 5 3z z z

2 2 2

2 2

7 8 5 3 7 8 5 3 7 4 2 5 3

7 2 2 5 3 14 2 5 3 14 5 3 2

70 6 70 6

z z z z z z z z z

z z z z z z z z z

z z z z

E.g.: Simplify 5 3 6 4x x

Using the distributive property we get

5 3 6 4 5 3 6 5 4 15 6 20 15 6 20x x x x x x x x x x x

E.g.: Simplify 9 5 3 7y y

Using FOIL, or the box method, or the happy rainbow method, etc. we get

2

9 5 3 7 9 3 9 7 5 3 5 7

27 63 15 35 27 63 15 35 63 8 15

y y y y y y

y y y y y y y y

E.g.: Simplify 2

3 11y

Using FOIL, or the box method, or the happy rainbow method, etc. we get

2

2

3 11 3 11 3 11

3 3 3 11 11 3 11 11

9 33 33 121

9 66 121

y y y

y y y y

y y y

y y

Do #’s 3 c, 4 a, b, 6 b, 8 c, d, 9, c, d, p. 212 text in your home work booklet.

Dividing Radical Expressions with Variables

Don’t forget about rationalizing the deniminator and giving the restrictions.

E.g.: Simplify 4

3

18

6

z

z

4 2 2 2

3 2

18 18 18 18 3 33 3 ; 0

666 6

z z z z z z z z zz z

zz z z z z z zz z z

E.g.: Simplify 315 7 10 54

5

y

y

Breaking the expression into two fractions gives

3 3

2

2

10 54 2 9 615 7 3 7

5 5

2 3 63 7

6 63 7

3 7 3 7 6 66 6; 0

y y y y

y y y y y y

y yy y

y y y y

y yy y

y y y y

y y yy yy

y y y

Do #’s 12, 4 c, d, 6 c, d, 10, 15, pp. 212-213 text in your home work booklet.

§4.5 & 4.6 Exploring and Solving Radical Equations (2 classes)

Read Goal pp. 214, 216 text.

Outcomes:

1. Identify strategies to solve radical equations involving square roots and cube roots. p. 215

2. Solve radical equations involving square roots and cube roots. pp. 216-221

3. Identify extraneous solutions. P. 217

Solving radical equations makes use of inverse operations.

Addition and subtraction are inverse operaions.

Multiplication and division are inverse operaions.

Squaring and taking the square root are inverse operations.

Cubing and taking the cube root are inverse operations.

Solving radical equations with square roots or cube roots often involves squaring or cubing both sides of

the equation. This process may introduce lead to solutions that do not work when substituted back into

the original equation. These solutions are called extraneous solutions. Therefore, when you solve by

squaring or cubing you have to verify each solution to ensure it is not extraneous. Extraneous roots can

also be identified from the restrictions on the variable(s)

E.g.: Solve 5 8y

Squaring both sides gives

2

25 8

5 64

64; 0

5

y

y

y y

Since we squared both sides we must check for extraneous roots (verify the solution).

LHS 5 64

564 8 RHS

So the solution is 64

5y .

E.g.: Solve 2 9x

Squaring both sides gives

2 2

2 9

2 81

83; 2

x

x

x x


LHS 83 2 81 9 RHS

So 83 is NOT a solution. It is an extraneous root. There is no solution to this equation. Do you see why?

E.g.: Solve 5 6 18x

2

2

5 6 6 18 6

5 12

5 12

5 144

149; 5

x

x

x

x

x x


LHS 149 5 6 144 6 12 6 18 RHS

So the solution is 149x .

E.g.: Solve 3 3 6z

Cubing both sides gives

3

33 3 6

3 216

3

z

z

3

z 216

3

72;z z

Since we cubed both sides we must check for extraneous roots (verify the solution).

33LHS 3 72 216 6 RHS

So the solution is 72z .

E.g.: Solve 3 7 34 3 2x

3

3

333

7 34 3 3 2 3

7 34 5

7 34 5

7 34 125

7 34 34 125 34

7 91

7

x

x

x

x

x

x

7

x 91

7

13x

Since we cubed both sides we must check for extraneous roots (verify the solution).

3 33LHS 7 13 34 3 91 34 3 125 3 5 3 2 RHS

So the solution is 13x .

Do #’s 2 c, d, 5, 6, 8 b, c, 15, pp. 222-224 text in your home work booklet.

Do #’s 1 b, d, 2 b, d, 3, 4, c, d, 5 c, d, 7, 9 b, d, 10, b, d, 11, p. 228 text in your home work booklet.

Problem Solving with Radicals

E.g.: The motion of a pendulum can be modeled using the formula 29.8

LT where T is the time (in

seconds) for one complete swing (over and back) and L is the length of the rope (in metres). If it takes a

pendulum 3.0s to make one complete swing, find the length of the rope.

2

2

2

2

2

2

3.0 29.8

3.0 29.8

9.0 49.8

9.0 9.8 4

9.0 9.8 4

4

L

L

L

L

24

L

2.2m L

The rope is about 2.2m long.

E.g.: A sphere has a surface area of 250.27m . Find the radius of the sphere. 2. . 4S A r

250.27 4

50.27 4

4

r

2

4

r

250.27

4

2.0m

r

r

The radius of the sphere is about 2.0m.

E.g.: A sphere has a volume of 3523.60m . Find the radius of the sphere. 34

3V r

3

3

3

3

4523.60

3

43 523.60 3

3

1570.80 4

1570.80 4

4 4

r

r

r

r

3

3 33

1570.80

4

1570.80

4

5.0m

r

r

r

The radius of the sphere is about 5.0m.

Do # 13, p. 223 text in your home work booklet.

Documents

Ch. 4 Notes - Mr. McDonald'smrmcdonaldshomepage.weebly.com/.../1/3/9/2/13921997/ch._4__radicals_.pdf · Math 2201 Unit 4: Radicals Read Learning Goals, p. 173 text. Ch. 4 Notes 4.1