Chapter 1RADICAL FUNCTIONS

Math Box• Suppose a and b are real numbers and n is a positive

integer not equal to 1 such that an = b then, a is the nth root of b.

Example: 25 = 32 2 is the 5th root of 32 33 = 27 3 is the cube root of 27

52 = 25 5 is the square root of 25

Rational Exponents: Its Roots

• If n is a POSITIVE ODD INTEGER and b is a REAL NUMBER, then b has exactly ONE REAL ROOT called principal nth root of b.

• If n is a POSITIVE EVEN INTEGER and b is a REAL NUMBER, then b has TWO NTH ROOTS (negative and positive)

• If n is EVEN POSITIVE INTEGER and b is a NEGATIVE NUMBER then b has NO REAL NTH ROOT.

Rational Exponent: Its Definition

bbb mnn

m

nm 11

Lets Apply the Definition!

832

1645 x27 3 3

2

Simplify the following Rational Expressions:

423

251 2

3

923

Activity 1:

Simplify the following Rational Exponents

RADICALS

For any real number a and b and all integers n>0

abn n is the index or orderb is the radicand√ is the radical sign

is the radical expressiona is the nth root of b

abn

Radical Expression

sRadicand Index

3 4x

5 35xyx8 5

Writing Rational Exponents form into Radical form

Rational Exponent Radical Form

b Base Radicand

n Denominator of the rational exponent Index or order

m Numerator of the rational exponent

Power of the whole radicand

Rewrite the following Rational Exponents toRadical Form

421

x7 21

732 x3 2 3

1

Activity B:

Rewrite the following Rational Exponents toRadical Form

Rewrite the following Radical Form toRational Exponents

3 5 35 x

4 2x xy4

Activity C:

Rewrite the following Radical Form to

Rational Exponents

LAWS OF RADICALS

Laws of Radicals1. When b ≠ 0 and n>1.

Example:

2. When b < 0 and n is even.Example:

3.Example:

bn nb

bn nb

bn bn

3 32 5 54

5 2 442

3 53 5 2

5

Laws of Radicals4. Example:

5. Example:

6.Example:

nnn baab

n

nn

ba

ba

mnn m bb

3 8x 125

365

3

83

3 5 3 4 2

Answer the following by applying theLaw of Radicals

1. 5.

2. 6.

3. 7.

4. 8.

3 53

6 26

5012

43

3

278

16

5 75

Simplification of Radicals

A radical expression is said to be simplified or in simplest form if:

• Case 1: The radicand has no factors whose indicated roots can still be taken.

• Case 2: The radicand does not contain a fraction.• Case 3: The denominator does not contain a radical

expression.• Case 4: The index or the order of the radical is in its

lowest form.

Case 1: The radicand has no factors whose indicated roots can still be taken.

yx45

3

4

16

121.2.3.

Case 2: The radicand does not contain a fraction

yx3

3

54411

.2.3.

Case 3: The denominator does not contain a radical expression

1.

2.

3.3 275523

x

Case 4: The index or the order of the radical is in its lowest form

1.

2.

3.

4. 1248

6 333

6

4

16

869

pn

zyxx

Before Class Activity

In your Math BookPage 7

Items 1-10

Operations of Radicals

Similar Radicals are radicals with the same indices and radicand when simplified.

Examples:

37,234,

532

,22,2,27

3,5,2333 xxxx

xxxx

Multiplication of Radicals

Multiplication of Radicals with the SAME INDICES.1. Multiply their radicands2. Multiply their numerical coefficients3. Retain the common indices4. Simplify the product

Multiplication of RadicalsExamples: 1.

2.

3.

4. 132132

634

4432

35

2 3

xx x

Its Your Turn!Warm-Up Practice

Activity APage 20

ODD Items Only

Multiplication of Radicals

Multiplication of Radicals with the DIFFERENT INDICES.1. Make their indices the same by transforming

them to a fractional exponent.2. Take the LCD of their fractional exponents.3. Transform the radical form.

Multiplication of RadicalsExamples: 1.

2. xx 22

233

3

Addition and Subtraction ofSimilar Radicals

Similar Radicals are radicals with the same indices and radicand when simplified.

Make each pair of radical SIMILAR

75,27

63,28

18,2

45,5

12,31.6.

2.7.

3.8.

4.9.

5. 10.

75,45

36,24

32,2

50,2

12,48

Addition and Subtraction ofSimilar Radicals

Examples: 1.

2.

3.

4.

5. 313

505823

352

252724

525453

333

xxx

xxx

Its Your Turn!Warm-Up Practice

Activity APage 13

Items 1-7

Divisions of Radicals

Quotient Rule:

n

n

nyx

yx

Divisions of RadicalsSimplify: 1.

2.251593

Divisions of RadicalsSimplify: 1.

2.2712188

Divisions of RadicalsSimplify: 1.

2.

3.250

832

630

2

3

xx

Rationalizing the Denominator

Rationalize: 1.

2.

3.

4. 23

1052632

55

Its Your Turn!Warm-Up Practice

Activity BPage 27

Items 1-8

Conjugate of a Denominator

• If is the denominator, the conjugate is .• If is the denominator, the conjugate is .

ba

ba

ba ba

Give the conjugate of each expression:

1.

2.

3. 12

35

13

3

Conjugate the denominator then multiply

1.

2.

3.123

3255213

3

x

Its Your Turn!Warm-Up Practice

Activity CPage 27

Items 1-6