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Unit I Chapter 4 – Radicals 1

Section 4.1: Mixed And Entire Radicals

(I) Square Roots

Determine the area of the square.

Determine the dimensions of the square given the area.

Principal and Secondary Square Roots

What number(s) when squared give 49?

In other words, what x-values are solutions to x2 = 49 ?

7 cm

7 cm

49 cm2

x

x

Objectives:

Principal and secondary square roots

Evaluating Square Roots

Evaluating Cube Roots

Simplifying Roots by Calculator

Terminology of a Radical

Mixed and Entire Radicals


The number 49 has two square roots, one positive and one negative.

•The positive square root such as √49 is called the __________________

•The negative square root such as −√49 is called the _________________

A negative number does not have a square root in the real number system.

(II) Evaluating Square Roots

There are a couple of methods to evaluate square roots:

Method 1: Simplifying Square Roots by Perfect Square Numbers

Perfect Square Numbers or Factors to remember:

22 = 4

32 = 9

42 = 16

52 = 25

62 = 36

72 = 49

82 = 64

92 = 81

102 = 100

112 = 121

122 = 144

132 = 169

142 = 196

152 = 225

Example: Simplify

(a) √36 (b) √12 (c) √48

Perfect Square Numbers or Factors


Method 2: Simplifying Square Roots by Prime Factorization

Remember: Prime numbers

•A whole number (except 0 and 1) that is divisible

by 1 and itself.

Example: 2, 3, 5, 7, 11, 13, 17, 19, ……

Simplify each square root by generating the prime factorization

of the radicand.

(a) √36 (b) √12 (c) √48

The square root of a non–perfect square is an irrational number.

(III) Cube Roots

Determine the volume of the square prism.


The volume of a Borg Cube attacking earth is 64 000 000 m3.

Determine the edge length of the cube.

All numbers (positive and negative) have only one cube root, denoted by 3

The cube root of a perfect cube is a rational number.

Example. Simplify √10003

The cube root of a non–perfect cube is an irrational number.

Example. √73

(IV) Evaluating Cube Roots

There are a couple of methods to evaluate square roots:

Method 1: Simplifying Cube Roots by Perfect Cube Numbers

Perfect Cube Numbers or Factors to remember:

23 = 8

33 = 27

43 = 64

53 = 125

63 = 216

73 = 343

83 = 512

Perfect Square Numbers or Factors


Example: Simplify

(a) √2163

(b) √1203

Method 2: Simplifying Cube Roots by Prime Factorization

Example: Simplify each cube root by generating the prime factorization

of the radicand.

(a) √643

(b) √1203

(c) √1083

(V) Simplifying Roots by Calculator

Example 1

Use a calculator to determine the exact value of the following.

(a) 64 (b) 25 (c) 3 729 (d) 3 27

Example 2

Without using a calculator, determine the exact value of the following.

(a) 100 (b) 24 (c) 3 125 (d) 3 72


(VI) Terminology of a Radical

200 is an example of an entire radical where the index is 2 and the radicand is 200

3 162 is an example of an entire radical where the index is 3 and the radicand is 162

32 and 3 54 are examples of mixed radicals

(VII) Mixed and Entire Radicals

Example 3

Express each of the following radicals as a mixed radical.

(a) 32 (b) 75 (c) 3 54 (d) 3 40

Example 4 Express each of the following mixed radicals as an entire radical.

(a) 33 (b) 52 (c) 3 24 (d) 3 43

Questions: P.182 – 183 #1a, b, d, e, #2, #4, #5, #6a, b, #10

#11,#12, #16a, #17


Section 4.2: Adding And Subtracting Radicals

(I) Review – Addition and Subtraction of Polynomials

Example: Determine the perimeter (in simplest form) of the

rectangle below.

When adding (or subtracting) polynomials only

like terms can combine.

Example: Simplify

2x2y – 4xy2 + 5x2y – 10xy2

5x2 + 8x

3x2 + 4x

same variable(s)

same exponent

Objectives:

Review Addition/Subtraction of Polynomials

Addition/Subtraction of Radicals


(II) Addition and Subtraction of Radicals

Example: Determine the perimeter (in simplest form) of the

rectangle below.

When adding (or subtracting) RADICALS only

like radicals

can combine.

Radicals may have to be simplified by considering perfect square or perfect

cube factors before they can be combined with other like radicals.

Example: Simplify 2√8 − 5√12 + 3√18 + √27

same index number

same radicand

5√3 + 8√2

3√3 + 4√2

Remember

only like radicals can combine


Example 1

Determine whether the set of radicals below are like or unlike terms.

(a) 18,25,22 (b) 32,32,33 (c) 50,45,5

Example 2

Add or subtract the following radicals and express the answer in simplest radical form.

(a) 2722 (b) 273934 (c) 542055


Example 3

Simplify the following and express the answer in simplest radical form.

(a) 8045255 (b) 98122

1502274

Example 4

Determine the perimeter of the following diagram.

28

63 63

112

Questions: P.188 – 190 #1, #2, #3, #4, #5, #6, #7, #9, #11,

#12, #14, #15, #16, #18


Section 4.3: Multiplying And Dividing Radicals

(I) Multiplying Monomials

Example: Determine the area for the rectangle below

5xy3

3x2y

Note: When monomials multiply

coefficients (numbers) multiply

add exponents when same variable multiples

Objectives:

Review Multiplying Monomials

Multiplying Radicals

Review of Rational and Irrational Numbers

Dividing Radicals (Rationalizing the Denominator)

Simplifying Radicals by Division


(II) Multiplying Radicals

Example: Determine the area for the rectangle below

The product of two mixed radicals

is equal to the product of the rational numbers times the product of the radicands.

ac x bd = abcd eg. 24 x 35 = 620

Example 1

Express each product in simplest radical form.

(a) 6 x 3 (b) 22 x 63 (c) 15 x 104

Note: When radicals multiply

rational numbers in front multiply

radicands (numbers underneath) multiply

simply resulting root if possible

5√6

3√2

Rational

numbers

Radicands


Example 2

Expand each expression and simplify.

(a) 823 (b) 332345 (c) 25332

(III) Rational and Irrational Numbers

A rational number

terminates or has a repeating decimal

Example: 2

An irrational number

•are non-terminating or non- repeating decimals

Example: √2 ≈ 1.414213562 …

Multiplying radicals with the same radicand will produce a ______________

Example: Multiply and simplify.

√2 𝑥 √2 =

Irrational numbers


(IV) Dividing Radicals (Rationalize the Denominator)

Determine the area for the rectangle below

Determine the length of the rectangle below

√8

√2

L

√2 Area = 4

Rationalizing the Denominator

•changing the denominator to a rational number

•𝐼𝑓 𝑎√𝑏

𝑐√𝑑 𝑡ℎ𝑒𝑛

𝑎√𝑏

𝑐√𝑑 𝑥

√𝑑

√𝑑


(V) Simplifying Radicals by Division

b

a

b

a Example: Simplify

2

6

b

a

d

c

bd

ac Example: Simplify

75

1415

Example 1

Express each of the following in simplest radical form.

(a) 10

120

(b) 63

486

(c) 3

3


Example 2

Rationalize the denominator and simplify.

(a) 63

85

(b) 32

12423

(c) 24

45125

Example 3

Determine the width, w, of the given rectangle.

Area = 340

l = 64

w

Questions: Pages 198–200, # 1, 2a, 4, 5, 6a, 8, 9a, 10, 13, 14, 16, 17, 19


INCLASS ASSIGNMENT REVIEW – RADICALS

(I) Simplifying Radicals

Express as a radical in simplest form.

(a) √63 (b) 3√24 (c) √483

(d) 2√543

(II) Entire and Mixed Radicals

Express as a mixed radical.

(a) √28 (b) √243

(c) √50 (d) √1283

Express as an entire radical.

(e) 3√5 (f) 5√3 (g) 2√43

(h) 5√23

(III) Addition and Subtraction of Radicals


(a) 2√3 + 3√6 + 5√3 − 8√6 (b) √32 + √75 − 2√48 − 2√50

(c) 3√18 − √45 + 1

2√8 + 3√20

(IV) Multiplication of Radicals


(a) 2√6 x 3√15 (b) 5√2 ( √27 + 3√12 − √24 )

(c) (3√2 − 4√6)(5√2 + √6) (d) (2√5 − √10)2


(V) Division of Radicals


(a) 12

√3 (b)

4√2

3√10 (c)

2√24 − 7√6

3√3 (d) (

2√8+ 4√18

4√5)

(VI) Perimeter and Area of Figures

(a) Determine the perimeter of the following trapezoid.

(b) Determine the area of the following rectangle.

(c) Determine the width, w, of the given rectangle.

12

27 27

75

52

83

Area = 2418 w

l = 39


SOLUTIONS

(I) (a) 3√7 (b) 6√6 (c) 2√63

(d) 6√23

(II) (a) 2√7 (b) 2√33

(c) 5√2 (d) 4√23

(e) √45 (f) √75 (g) √323

(h) √2503

(III) (a) 7√3 − 5√6 (b) −6√2 − 3√3 (c) 10√2 + 3√5

(IV) (a) 18√10 (b) 45√6 − 20√3 (c) 6 − 34√3 (d) 30 − 20√2

(V) (a) 4√3 (b) 4√5

15 (c) −√2 (d)

4√10

5

(VI) (a) 13√3 (b) 12√10 (c) w = 4√2


Section 4.4: Simplifying Algebraic Expressions Involving Radicals

(I) Radicals that contain variables.

Example: Simplify.

(a) √𝑥2 (b) √𝑥3

(c) √𝑥4 (d) √𝑥5

(e) √𝑥6 (f) √4𝑥4

(g) √12𝑥2 (h) √18𝑥3

Objectives:

Radicals the Contain Variables

Adding/Subtracting Radicals that Contain Variables

Multiplying Radicals that Contain Variables

Dividing Radicals that Contain Variables

Simplifying Radicals by Division


Example: Express as an entire radical.

(a) 3𝑥√2 (b) 2𝑥 √3𝑥3

(II) Adding/Subtracting radicals that contain variables

Only like radicals can combine when adding or subtracting.

Example. Simplify

(a) 22 52 xx (b) 33 823 xx

(c) 4185 x (d) 52 277 mm


(III) Multiplying radicals that contain variables

Multiply the following algebraic expressions involving radicals.

(a) xxx 2263 2 (b) xxx 324 (c) 22 5432 xx

(IV) Dividing radicals that contain variables

Example: Simplify

(a) x

x

3

12 3

(b) x

xx

2

53 3

(c) x

xx

3

6223 3

Questions: Pages 212–213, # 3, 4, 6a, d, 5, 6a, 8b, c, d, 9a, d, 10b, c, 12a, d, 15


Section 4.6: Solving Radical Equations

(I) Solving radical equations

Example 1 Solve for x.

(a) 5x (b) 43 x (c) 63 x

(d) 223 x (e) 1223 x

Objectives:

Solving Radical Equations

Applications of Radical Equations

To solve radical equations:

(i) May need to isolate the radical first

(ii) Square both sides (√ )2

= ( )2 to eliminate a square root

OR cube both sides (√3

)3

= ( )3 to eliminate a cube root


Example 2

Solve the following radical equations.

(a) 45 x (b) 383 x

(c) 10732 x (d) 1220102 x


Example 3

The forward and backward motion of a swing can be modeled using the formula

8.92

LT

where T represents the time, in seconds, for a swing to return to its original position and L

represents the length of the chain, in meters, supporting the swing.

If it takes 2 seconds for the swing to return to its original position then what is the length of the

chain supporting the swing?

Questions: Pages 222–224, # 1, 2, 4, 5, 8, 11, 12, 13, 17


RADICALS TEST REVIEW

1. Simplify the following.

(a) 3 6481 (b) 25273 (c) 75 (d) 3 24 (e) 398x (f)

3 54

2. Express as an entire radical.

(a) 73 (b) 3 32 (c) 24x (d) 3 45x

3. Perform the operations indicated and express the answer in simplest radical form.

(a) 737974 (b) 274812 (c) 85454322

1202

4. Determine the perimeter of the following diagrams.

(a) (b)

5. Perform the operations indicated and express the answer in simplest radical form.

(a) 3483 x (b) 56142 (c) 24632 (d) 235

(e) 22233 (f) 5825 (g) 272232334

6. Determine the area of the following diagrams.

(a) (b)

20

45 45

125

28

63

22

65

34

34


7. Rationalize the denominator.

(a) 5

60

(b) 33

5415

(c) 2

8

(d) 58

82

(e) 33

24268 (f)

24

452202

8. Determine the width of the given rectangles.

(a) (b)

9. Simplify the following algebraic expressions involving radicals.

(a) 694 x (b) 2272 x (c) 4503 xx (d) xxx 3265 2

(e) xxx 243 5 (f) 22 2536 xx

(g)

x

x

3

12 5

(h)

x

xx

52

153 3

(i)

x

xxx

3

272124 3

10. Solve the following radical equations.

(a) 4x

(b) 33 x

(c) 93 x

(d) 822 3 x

(e) 1823 x

(f) 53 x

(g)

243 x (h) 14952 x

(i) 64233 x

(j)

10482 x

(k)

1711153 x

Area = 5416

l = 34

w Area = 9627

l = 29

w


SOLUTIONS

1. (a) 5

(b) – 2

(c) 35

(d)

3 32

(e) xx 27

(f)

3 23

2. (a) 63

(b)

3 24

(c)

232x

(d) 3 3500x

3. (a) 72 (b) 33

(c) 21258

4. (a) 710 (b) 513

5. (a)

624 (b)

76

(c) 218 (d)

1528 (e)

61235

(f)

1039

(g)

6654

6. (a) 320 (b) 48

7. (a)

32

(b) 215

(c) 24

(d) 10

10 (e)

3

24 (f)

4

10

8. (a) 212 (b) 312

9. (a) 312x (b) 36x (c) 215 3x (d) xx 230 2

(e) xx 612 3

(f) 152412 2 xx

(g) 22x

(h) 2

33 2x

(i)

26x

10. (a) 16x

(b) 3 27x

(c) 27x

(d) 32x

(e) 18x (f) 28x

(g)

12x

(h) 8x

(i) 6x

(j)

22x

(k)

7x

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