chapter 9 section 1 evaluating square roots. learning objective 1.evaluate square roots of real...

Chapter 9 Section 1

Evaluating Square Roots

Learning Objective1. Evaluate Square Roots of real numbers

2. Recognize that not all square roots represents real numbers

3. Determine whether the square root of a real number is rational or irrational

4. Write square roots as exponential expressions

Key Vocabulary

• square root • principal square root• radical sign• radicand• radical expression• index

rootimaginaryperfect squareperfect square

factorrational numberirrational number

Evaluate Square Roots of Real Numbers

• positive or principal square root uses the to indicate a positive square root if

• “The square root of a”

• Negative square roots are indicated by

•

a

a

a b 2b a

a

a imaginary not a real number

Evaluate Square Roots of Real Numbers

• radical sign is

• radicand is the number under the radical sign

• radical expression is the entire expression

• Index tells the root of the expression and the squared root index are not written

• All squares of a nonzero real number must be positive

a

2 x x

Evaluate Square Roots of Real Numbers

• square root is the reverse process of squaring a numberExample : because 72 = (7)(7) = 49

• The is 0, written

49 7

0 00

Evaluate Square Roots of Real Numbers

• Examples: 2

2

2

2

2

2

49 7 because 7 (7)(7) 49

0 0 because 0 (0)(0) 0

9 3 because 3 (3)(3) 9

900 30 because 30 (30)(30) 900

49 7 because (-7) ( 7)( 7) 49

64 8 because (-8)

2

( 8)( 8) 64

36 6 because (-6) ( 6)( 6) 36

Evaluate Square Roots of Real Numbers

• Examples:2

2

2

36 6 6 6 6 36 because

121 11 11 11 11 121

25 5 5 5 5 25 because

81 6 9 9 9 81

225 15 15 15 15 225 because

49 7 7 7 7 49

Negative Square Roots• Negative square roots are not real numbers

• How do we know that the square of any nonzero real number must be a positive number?

Example:

2

2

16 4 ( - 4) ( 4)( 4) 16

16 ( 4)( 4) 16 16

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

36 ( 6)( 6) 36

real because

imaginary not a real number because not

real because

imaginary not a real number because no

36

121 ( 11)( 11) 121 121

t

imaginary not a real number because not

Perfect Squares

• The numbers 1, 4, 9, 16, 25, 36, 49, … are perfect squares because each number is a square of a natural number.

2

2

2

2

2

1 1 because 1 (1)(1) 1

4 2 because 2 (2)(2) 4

9 3 because 3 (3)(3) 9

16 4 because 4 (4)(4) 16

25 5 because 5 (5)(5) 25

See page 536 for a list of the first 20 perfect squares.

Natural numbers 1 2 3 4 5Square Natural number 12 22 32 42 52

Perfect squares 1 4 9 16 25

Rational Numbers• A rational number can be written as

a and b are integers and b ≠ 0

• Rational numbers written as a decimal are either terminating or repeating. ½ = .5 or ⅓ = .333…

• Round you answers two decimal place and use the approximately equal sign ≈

ab

Rational Numbers• The square root of every perfect square is also a rational

number.

225 15 rational number

64 = 8 rational number

361 = 19 rational number

0 0 rational number

9 3 rational number

900 30 rational number

64 8 rational number

25 5 rational number

81 9

125 15 rational number

49 7

Irrational Numbers• A irrational number is any number that is not rational and are

non-terminating and non-repeating decimals.

• The square root of non perfect square are irrational number

38 6.164414 irrational number

74 8.6023252 irrational number

320 17.888543 irrational number

31 5.567764 irrational number

43 6.557438 irrational number

Writing a Square Root in Exponential Form

1/ 22

1/ 2 2 1/ 22 1

1/ 2 2 1/ 22 1

:

x

5 5 5 5

x x x

Because

x x x

Reviewing the rules for exponents in chapter 4 section 1 and 2 may be helpful.

Writing a Square Root in Exponential Form

1

2

1

2

1

2

12 2 2

13 3 3 3 2

Examples: Write a square root in exponential form

15 15

71 71

11 11

17ab 17

59 59

x x

ab

x y x y

1

2

1

2

12 2 2

13 3 2

15 3 5 3 2

39 (39 )

3 3

5x = 5x

7 7

22 22

x x

x y x y

m n m n

Review of Rules for Exponents

m n m nx x x

0 , xxx

x nmn

m

1 0 x

Product Rule:

Quotient Rule:

Zero Exponent Rule:

Review of Rules for Exponents

))(( nmnm xx

0 0, ,

ybyb

xa

by

axmm

mmm

0 ,x

1 m xx m

Power Rule:

Expanded Power Rule:

Negative Exponent Rule:

Remember

• The square root is the opposite or reverse process of squaring.

• “Square of a number”

• “Multiply the number by itself”

• Every real number greater than 0 has two squares one positive and one negative

• Square roots of negative numbers are not real numbers. They are imaginary numbers

Remember

• The results of a square root is always nonnegative.

• The result is only rational if the radicand is a perfect square

• The radical form is the exact value.

• Calculators only give approximate values for irrational numbers. We use the ≈ sign for these values.

Remember

• You should try to memorize as many perfect squares as possible to help with simplifying in the next section.

• Review of factoring may also be helpful for simplifying in the next section.

• Extra practice may be helpful to remember the difference between and 5 5

HOMEWORK 9.1

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