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Evaluating Square Roots
Chapter 9 Section 1
Learning Objective
1. 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
 root
 imaginary
 perfect square
 perfect square
factor
 rational number
 irrational number
Evaluate Square Roots of Real
Numbers
• positive or principal square root uses the a to indicate a
positive square root a  b if b 2  a
•
a “The square root of a”
• Negative square roots are indicated by  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
a
• Index tells the root of the expression and the squared root
index are not written 2 x  x
• All squares of a nonzero real number must be positive
Evaluate Square Roots of Real
Numbers
• square root is the reverse process of squaring a number
Example : 49  7 because 72 = (7)(7) = 49
• The 0 is 0, written
0 0
Evaluate Square Roots of Real
Numbers
• Examples:
49  7
7 2  (7)(7)  49
because
0 0
because
02  (0)(0)  0
9 3
because
32  (3)(3)  9
900  30
because
302  (30)(30)  900
 49  7 because
(-7) 2  (7)(7)  49
 64  8 because
(-8) 2  (8)(8)  64
 36  6 because
(-6) 2  (6)(6)  36
Evaluate Square Roots of Real
Numbers
• Examples:
36
6

121 11
2
 6   6  6  36
      
 11   11  11  121
because
25 5

because
81 6
2
 5   5  5  25
      
 9   9  9  81
225
15


because
49
7
2
 15   15  15  225
         
 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:
 16  4
16
real because ( - 4) 2  ( 4)( 4)  16
imaginary not a real number because ( 4)( 4)  16 not  16
 36  6 real because ( - 6) 2  ( 6)( 6)  36
36
121
imaginary not a real number because ( 6)( 6)  36 not  36
imaginary not a real number because ( 11)( 11)  121 not  121
Perfect Squares
• The numbers 1, 4, 9, 16, 25, 36, 49, … are perfect squares
because each number is a square of a natural number.
1 1
because
12  (1)(1)  1
42
because
22  (2)(2)  4
9  3 because
32  (3)(3)  9
16  4
because
42  (4)(4)  16
25  5 because
52  (5)(5)  25
See page 536 for a list
of the first 20 perfect
squares.
Natural numbers
1 2 3 4 5
Square 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
a
b
• 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 ≈
Rational Numbers
• The square root of every perfect square is also a rational
number.
225  15
64 = 8
361 = 19
0 0
9 3
rational number
900  30
rational number
 64  8
rational number
25 5

81 9
rational number
rational number
rational number
rational number
rational number

125
15

49
7
rational number
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
x xx
Reviewing the rules for exponents
in chapter 4 section 1 and 2 may
be helpful.
1/ 2
2
Because :
x

2 1/ 2
5 
2 1/ 2
 2 1/ 2 
 x1  x
 2 1/ 2 
5 5
x
5
1
Writing a Square Root in Exponential Form
Examples: Write a square root in exponential form
15  15 
39 x  (39 x)
1
2
71   71
3   3
1
2
11x  11x 
17ab  17 ab
2
2
1
2 2

59 x y   59 x y
3
3
1
2
5x =  5x
1
2
3
1
2
1
2 2

7x y  7x y 
3
1
3 2

3
1
2
1
5 3 2
22m n   22m n
5 3

Review of Rules for Exponents
Product Rule:
x x x
m
n
mn
m
Quotient Rule:
Zero Exponent Rule:
x
mn
x , x0
n
x
x 1
0
Review of Rules for Exponents
Power Rule:
x 
m n
x
( m)( n )
m
Expanded Power Rule:
Negative Exponent Rule:
 ax 
am xm
   m m , b  0, y  0
b y
 by 
x
m
1
 m , x0
x
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  5 and 5
HOMEWORK 9.1
Page 539 - 540
# 13, 21, 23, 31, 33, 37, 41, 65, 71, 74