11.1 and 11.2 Radicals

• Goal(s):

1. To find the square roots of perfect squares, perfect square radicands and estimate the roots of irrational numbers

2. Determine whether a number is rational or irrational

3. Determine acceptable replacements for radicands

Square Roots• When we raise a number to the second

power, we have “squared” the number.• Sometimes we need to find the number

that was squared. That process is called “finding the square root of a number”.

• Every positive number has both positive and negative square roots.

an25 5 5d -Radical Sign

25 25

25 5

“Principal” Square Root is the positive 5.

Simplify: (Answer is only the principal root)

81 9

64 -8 525 Negative square Root

Because (9)2 = 81

36(6)2 -36(-6)2 -36

Not possible

Perfect Square Radicands

|| 2 xx

Simplify: 21x

|1| x

Simplify: 1682 xx

|4| x

24x

Simplify: 225x

|5| x

|x|5or

Simplify:2

4

1x

x2

1 1

2x

• Comes from the word “ratio”

• Any number that can be expressed as the ratio of two integers

Rational Numbers

71

7,yes

1.310

13,yes

0.33331

,3

yes

Irrational Numbers• Cannot be written as the ratio of two

integers.• Decimal never ends and does not repeat.• Examples of irrational numbers:

2

0.4545 25

Identify the number as irrational or rational

35 36The square roots of most whole numbers are irrational. Only the perfect squares (0, 1, 4, 9,

16, 25, 36, etc.) have rational square roots.

6

1

Identify the rational number:

. 48

. 49

. 50

. 51

A

B

C

D

49 7

Real Numbers: All the rational and all the Irrational numbers.

Real Numbers

Rational Numbers

Irrational Numbers

25 is a real number.N T- O

2

95

0

7

25 25

25 25

Approximate the value of 29

25 5

36 6

29 5.3

Approximate the value of 12

9 3

16 4

12 3.4

Approximate the value of 75

64 8

81 9

75 8.7

Approximate the value of 45

36 6

49 7

45 6.8

Find without a calculator: 225

100 10

400 20

211 121212 144

213 169

15

Find without a calculator: 289

100 10

400 20

213 217

17

Radical Expressions(an expression written under a radical)

18 x 92 x

2

72 xRadicand

(the expression written under the radical)

The radicand must be positive!

Evaluate the expression for x = 5.Is the result a real number?

4x

945

3

Evaluate the expression for x = 2.Is the result a real number?

6x

462

number reala not

Determine the values of x that make the expression a real number.

12 x012 x

2

1x

12 x

Determine the values of x that make the expression a real number.

32 x

032 x

onsManySoluti

32 x

Homework

• Work book Page 38 ( 2-32) even