SPECIAL FRACTION EXPONENT:n

1

The exponent is most often used in the power of

monomials.Examples: Do you notice any other type of mathematical symbols that these special fraction exponents represent?

3

13x 2

1423 x 4

1482 y

2

129 3

1636 x 2

1225 z

n

1

Radicals:

Index

Steps for Simplifying Square Roots

1. Prime Factorization: Factor the Radicand Completely

2. Write the base of all perfect squares (PAIRS) outside of the

radical as product

3. Everything else (SINGLES) stays under the radical as a

product.

Note: The square root or ½ exponent is the most common radical and does not need to have the index written.

Special Fraction Exponents, , are more commonly known as radicals in which the N value represents the root or index of the radical.

Na1

nn aa 1 Radical Symbol

Radicand

aa 2

1

Root Properties:

[1] 00 n

[2] An EVEN index (n), cannot take negative radicands.

answer real-non 4 [3] An ODD index (n), can take both positive and

negative radicands. Roots are the same sign as the radicand.

3 273 3 273

n a

2 4

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]

Simplifying Square Roots: Part 1

12[A]32[B]

436x[E]718 y[C]

Simplifying Square Roots: Part 2

12610 zyx[B]

91417 cba[D]

10412 yx[A]

181081 qp[B]

12730 yx[C]

151749 nm[D]

Radicals Classwork: Additional Practice 12428 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 62yx 12516 yx xyx 52 58

yyx 7211 29 57 yx 53 62yx

1462yx yyx 142 85 xyx 7858

yyx 6215 yyx 33 26 yyx 579

Simplifying Any RootSame General Steps: Take out only groups of size n (the index) for the same base from the radical. These groups are called perfect roots.

Example 1

a] 3 54 c] 3 40

rootcubetripletsa :3

rootofgroupsa th4:44 rootofgroupsa th5:55

b] 4 48

Example 2A] 3 11940 yx B] 3 1012250 yx

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

[E] 416x [F] 43 )5( x

Applications Using Roots

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

g

LT 2 seconds 36.1

32

5.12

T

[B] The distance D in miles from an observer to the horizon over flat land or water can be estimated by the formula below, where h is the height in feet of observation. How far 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: “Inside to Inside and Outside to Outside”

Multiplying Radical Expressions: Distribute and FOIL

[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

Foil METHOD PRACITCE

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

mmm 7535

mm 1235

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

8101215 xxx

8215 xx

ADD and SUBTRACT radical expressions 1)Find common radicand (simplify)2)Combine like terms (outsides only)

d] 3273122

b] 2218385

2229210

23

a] 552322 [c] 342326

3423

e] 3218283 f] 827122

2237

223334

5525

312

33934

28

242626