chapter 9 section 1 evaluating square roots. learning objective 1.evaluate square roots of real...
TRANSCRIPT
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
Page 539 - 540
# 13, 21, 23, 31, 33, 37, 41, 65, 71, 74