Introduction to Radicals

If b 2 = a, then b is a square root of a.

Meaning Positive

Square Root

Negative

Square Root

The positive and negative square

roots

Symbol

Example

39 39 39

Radical Expressions

Finding a root of a number is the inverse operation of raising a number to a power.

This symbol is the radical or the radical sign

n aindex

radical sign

radicand

The expression under the radical sign is the radicand.

The index defines the root to be taken.

• square root: one of two equal factors of a given number. The radicand is like the “area” of a square and the simplified answer is the length of the side of the squares.

• Principal square root: the positive square root of a number; the principal square root of 9 is 3.

• negative square root: the negative square root of 9 is –3 and is shown like

• radical: the symbol which is read “the square root of a” is called a radical.

• radicand: the number or expression inside a radical symbol --- 3 is the radicand.

• perfect square: a number that is the square of an integer. 1, 4, 9, 16, 25, 36, etc… are perfect squares.

39

39

3

Square Roots

If a is a positive number, then

a is the positive (principal) square root of a and

100

a is the negative square root of a.

A square root of any positive number has two roots – one is positive and the other is negative.

Examples:

10

25

49

5

7

11 36 6

9 non-real # 81.0 9.0

What does the following symbol represent?

The symbol represents the positive or principal root of a number.

4 5xyWhat is the radicand of the expression ?

5xy

What does the following symbol represent?

The symbol represents the negative root of a number.

3 525 yxWhat is the index of the expression ?

3

What numbers are perfect squares?

1 • 1 = 12 • 2 = 43 • 3 = 9

4 • 4 = 16 5 • 5 = 25 6 • 6 = 36

49, 64, 81, 100, 121, 144, ...

Perfect Squares

1

4

916

253649

64

81

100121

144169196

225

256

324

400

625

289

4

16

25

100

144

= 2

= 4

= 5

= 10

= 12

Simplifying Radicals

Simplifying Radical Expressions

100 4 25

36 4 9

326

Product Property for Radicals

ab a b

10 2 5 36 4 9

Simplifying Radical Expressions

• A radical has been simplified when its radicand contains no perfect square factors.

• Test to see if it can be divided by 4, then 9, then 25, then 49, etc.

• Sometimes factoring the radicand using the “tree” is helpful.

Product Property for Radicals

50 25 2 5 2

14 7x x

8

20

32

75

40

=

= =

=

=

2*4

5*4

2*16

3*25

10*4

=

=

=

=

=

22

52

24

35

102

Perfect Square Factor * Other Factor

LE

AV

E I

N R

AD

ICA

L F

OR

M

48

80

50

125

450

=

= =

=

=

3*16

5*16

2*25

5*25

2*225

=

=

=

=

=

34

54

25

55

215

Perfect Square Factor * Other Factor

LE

AV

E I

N R

AD

ICA

L F

OR

M

Steps to Simplify Radicals:

1. Try to divide the radicand into a perfect square for numbers

2. If there is an exponent make it even by using rules of exponents

3. Separate the factors to its own square root

4. Simplify

Simplify:12x

6x

26x

Square root of a variable to an even power = the

variable to one-half the power.

Simplify:88y

44y

Square root of a variable to an even power = the

variable to one-half the power.

Simplify:13x

xx6

12x x

112xx

Simplify:750y

35 2y y

625 2y y

Simplify

1. .

2. .

3. .

4. .

2 18

72

3 8

6 236 2

Simplify 369x

1. 3x6

2. 3x18

3. 9x6

4. 9x18

+To combine radicals: combine the coefficients of like radicals

Simplify each expression

737576 78

62747365 7763

Simplify each expression: Simplify each radical first and then combine.

323502

22

212210

24*325*2

2*1632*252

Simplify each expression: Simplify each radical first and then combine.

485273

329

32039

34*533*3

3*1653*93

18

288

75

24

72

=

= =

=

=

=

=

=

=

=

Perfect Square Factor * Other Factor

LE

AV

E I

N R

AD

ICA

L F

OR

M

Simplify each expression

636556

547243

32782

Simplify each expression

20556

32718

6367282

Homework radicals 1

• Complete problems 1-24 EVEN from worksheet

*To multiply radicals: multiply the coefficients and then multiply the radicands and then simplify the remaining radicals.

35*5 175 7*25 75

Multiply and then simplify

73*82 566 14*46

142*6 1412

204*52 1008 8010*8

2

5 5*5 25 5

2

7 7*7 49 7

2

8 8*8 64 8

2

x xx * 2x x

To divide radicals: divide the coefficients, divide the radicands if possible, and rationalize the denominator so that no radical remains in the denominator

7

56 8 2*4 22

7

6This cannot be

divided which leaves the radical in the

denominator. We do not leave radicals in the denominator. So

we need to rationalize by multiplying the

fraction by something so we can eliminate

the radical in the denominator.

7

7*

7

6

49

42

7

42

42 cannot be simplified, so we are

finished.

This can be divided which leaves the

radical in the denominator. We do not leave radicals in the denominator. So

we need to rationalize by multiplying the

fraction by something so we can eliminate

the radical in the denominator.

10

5

2

2*

2

1

2

2

This cannot be divided which leaves

the radical in the denominator. We do not leave radicals in the denominator. So

we need to rationalize by multiplying the

fraction by something so we can eliminate

the radical in the denominator.

12

3

3

3*

12

3

36

33

6

33

2

3Reduce

the fraction.

2X

6Y

264 YXP

244 YX

10825 DC

= X

= Y3

= P2X3Y

= 2X2Y

= 5C4D5

3X

XX

=

=

XX *2

YY 45Y

=

= YY 2

Homework:

worksheet --- Non-Perfect Squares (#1-12)

Classwork:

Packet in Yellow Folder under the desk --- 2nd page

Homework radicals 2

• Complete problems 1-15 from worksheet.