Simplifying Radicals

n aIndexRadical

Radicand

Steps for Simplifying Square Roots 1. Factor the Radicand Completely or until you find a perfect root

2. Take out perfect roots (look for pairs)

Note: With square roots the index isnot written

3. Everything else (no pairs) stays under the radical

Root Properties:

[1] 00 n

[2] If you have an even index, you cannot take roots of negative numbers. Roots will be positive.

answer real-non 4

[3] If you have an odd index, you can take the rootsof both positive and negative numbers. Roots may beboth positive and negative

3 273 3 273

General Notes:

[1] 416 4 is the principal root

[3] 416 ±4 indicates both primary and secondary roots

[2] 416 – 4 is the secondary root(opposite of the principal root)

24[C] 120[D]

Example 1

12[A] 32[B]

Example 2: Simplify84 yx[A] 12610 zyx[B]

12137 zyx[C] 91417 zyx[D]

Example 3:10412 yx[A] 71018 yx[B]

12730 yx[C] 151749 yx[D]

Radicals CW12428 yx[1] 2410256 yx[2]

101720 yx[3] 154121 yx[4]

1014162 yx[5] 12445 yx[6]

12414 yx[7] 171056 yx[8]

101778 yx[9] 134225 yx[10]

51227 yx[11] 111481 yx[12]

72 62yx12516 yx

xyx 52 58 yyx 7211

29 57 yx 53 62yx

1462yx yyx 142 85

xyx 7858 yyx 6215

yyx 33 26 yyx 579

Radicals

Simplifying Cube Roots (and beyond)

1. Factor the radicand completely

2. Take out perfect roots (triples)

Example 1

a] 3 54 b] 3 24

Example 2

a] 3 11940 yx b] 3 1012250 yx

Example 3 Finding Roots

[A] 416x [B] 43 )5( x

[C] 5 1510243 ba [D] 3 211264 ba

Example 4 Applications Using Roots

[A] The time T in seconds that it takes a pendulum to make acomplete swing back and forth is given by the formula below, where L is the length of the pendulum in feet andg is the acceleration due to gravity. Find T for a 1.5 footpendulum. Round to the nearest 100th and g = 32 ft/sec2.

g

LT 2

seconds 36.132

5.12

T

Example 5 Applications Using Roots

[B] The distance D in miles from an observer to the horizonover flat land or water can be estimated by the formulabelow, where h is the height in feet of observation. Howfar is the horizon for a person whose eyes are at 6 feet?Round to the nearest 100th.

hD 23.1

miles 01.3

623.1

D

Simplifying Radicals

Example 1 Multiplying Radical Expressions

[A]

3253 [C] [D] 3325

327 [B] 235

5215

6325315

1. Multiply radicand by radicand

2. If it’s not underneath the radical then do not multiply, write together (ex: )32

1525336

Example 2 Foil

a] )4)(2( xx b] )7)(5( mm

mmm 7535

mm 1235

c] )43)(26( xx d] )45)(23( xx

8101215 xxx

8215 xx

Example 3 Simplify Sums / Differences•Find common radicand•Combine like terms

a] 3273122 b] 2218385

2229210

23

Example 4 Adding / Subtracting Roots

[A] 552322 [B] 342326

3423

[C] 3218283 [D] 827122

2237

223334

Conjugate: Value that is multiplied to a radical expressionThat clears the radical.

Rationalizing: Multiplying the denominator of a fraction by its conjugate.

Example 1 Rationalizing Square Roots

[A]3

5 [B]5

7

5

5

5

57

Example 2 Rationalizing Square Roots Cont’

[A]3

5

x

[B]5

7

x

3

7

x

x

Example 3 Rationalizing Cube Roots

[A] 32

27

x[B] 3

7

3

x

[C] 34

125

x[D] 3

10

5

x

Example 4 Tougher Rationalizing

[A]yx3

5 [B] 3108

5

yx

[1] [2]

[3] [4]

[5] [6]

[7] [8]

[9] [10]

[11] [12]

Simplifying Radicals

237

5

1

453

3213 1515

7

3

3y

x5

3

x

37

5

x3

14

10

y

3114

2

yx5

15

xy

Simplifying Radicals

Binomial Conjugate: Binomial quantity that turns the expressioninto a difference of squares.

y2 y2 y4 x5 x5 x25

Example 1 Binomials Conjugates

[A]35

2

[B]

51

7

4

577

51

577

51

51

Rational Exponents Property: n

mn m xx

Example 1: Rational to Radical Form

A] 3

2

x B] 5

4

x C] 2

1

x

3 2x5 4x x

Radicals

Example 2: Radical to Rational Form

A] 3 5x B] C]

3

5

x

7 9x 5x

7

9

x 2

5

x

Radicals CW

3 5x

Write in rational form.

1. 2. 5 7x 3. 5x 4. 7 3x

3

1

x5. 6. 5

2

x 7. 2

5

x 8. 7

3

x

Write in radical form.

Radicals

Radical Equation Equation with a variable under the radical sign

Extraneous Solutions Extra solutions that do not satisfy equation

Radical Equation Steps[1] Isolate the radical term (if two, the more complex)

[2] Square, Cube, Fourth, etc. Both Sides

[3] Solve and check for extraneous solutions

Example 1 Solving Radical Equations Algebraically

[A] 421 x [B] 512 y

62 y

362 y

38y

Example 1

[C] 1551 x [D] 713 y

101 x1001x

101x

83 y

643 y

61y

Radicals CW

171122 x

4106 x

1711153 x

Solve Algebraically.

9. 10.4125 x

11. 12.

165 x2565 x251x

146 x1966 x190x

622 x3622 x382 x

19x

6153 x215 x415 x55 x1x

Radicals CW

95124 x4124 x

162147 x

Solve Algebraically.

13. 14.

112 x112 x02 x0x

14147 x

214 x414 x34 x

4

3x

Radicals CW

12121 xx

Solve Algebraically.

15. 16.475 xx

No Solution x = 4

Example 2 Solving Graphically

[A] 1513 xx [B] xx 2332

x = ½

Example 2 Continued

[C] 12121 yy [D] 161 xx

Y = 4 x = 3

Example 3 No Solutions

[A] xx 315 [B] 423 xx

x = Ø

Example 4 Misc. Equations

[A] 84 2 xx [B] 373 xx

x = 3

x = -1, -2