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Chapter Chapter 88Section Section 11
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Evaluating Roots
Find square roots.Decide whether a given root is rational, irrational, or not a real number.Find decimal approximations for irrational square roots.Use the Pythagorean formula.Use the distance formula.Find cube, fourth, and other roots.
11
44
33
22
66
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8.18.18.18.1
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 11
Find square roots.
Slide 8.1 - 3
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Find square roots.
When squaring a number, multiply the number by itself. To find the square root of a number, find a number that when multiplied by itself, results in the given number. The number a is called a square root of the number a
2.
Slide 8.1 - 4
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
The positive or principal square root of a number is written with
the symbol .
Find square roots. (cont’d)
Slide 8.1 - 5
0 0
a
Radical Sign Radicand
The symbol , is called a radical sign, always represents the
positive square root (except that ). The number inside the
radical sign is called the radicand, and the entire expression—radical
sign and radicand—is called a radical.
The symbol – is used for the negative square root of a number.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Find square roots. (cont’d)
Slide 8.1 - 6
The statement is incorrect. It says, in part, that a positive number equals a negative number.
9 3
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 1
Find all square roots of 64.
Solution:
Finding All Square Roots of a Number
Slide 8.1 - 7
Positive Square Root
Negative Square Root
64 8
64 8
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Find each square root.
EXAMPLE 2
Solution:
Finding Square Roots
Slide 8.1 - 8
169
225
13
15
25
64
25
64 5
8
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 3 Squaring Radical Expressions
Slide 8.1 - 9
Find the square of each radical expression.
Solution:
17 2
17 17
29 2
29 29
22 3x 222 3x 22 3x
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 22
Decide whether a given root is rational, irrational, or not a real number.
Slide 8.1 - 10
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Deciding whether a given root is rational, irrational, or not a real number.
Slide 8.1 - 11
All numbers with square roots that are rational are called perfect squares.
Perfect Squares Rational Square Roots
25
144
4
9
25 5
144 12
4 2
9 3
A number that is not a perfect square has a square root that is irrational. Many square roots of integers are irrational.
Not every number has a real number square root. The square of a real number can never be negative. Therefore, is not a real number.
-36
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 4Identifying Types of Square Roots
Slide 8.1 - 12
Tell whether each square root is rational, irrational, or not a real number.
27 irrational
36 26 rational
27 not a real number
Solution:
Not all irrational numbers are square roots of integers. For example (approx. 3.14159) is a irrational number that is not an square root of an integer.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 33
Slide 8.1 - 13
Find decimal approximations for irrational square roots.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Find decimal approximations for irrational square roots.
Slide 8.1 - 14
A calculator can be used to find a decimal approximation even if a number is irrational.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 5 Approximating Irrational Square Roots
Slide 8.1 - 15
Find a decimal approximation for each square root. Round answers to the nearest thousandth.
Solution:
190 13.784048 13.784
99 9.9498743 9.950
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 44
Slide 8.1 - 16
Use the Pythagorean formula.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Many applications of square roots require the use of the Pythagorean formula.
If c is the length of the hypotenuse of a right triangle, and a and b are the lengths of the two legs, then
Slide 8.1 - 17
Use the Pythagorean formula.
2 2 2.a b c
Be careful not to make the common mistake thinking that
equals .
2 2a b
a b
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 6
2 2 213 15a 2 169 225a
Using the Pythagorean Formula
Slide 8.1 - 18
7, 24a b
Find the length of the unknown side in each right triangle.
Give any decimal approximations to the nearest thousandth.
15, 13c b
118
?
2 2 27 24 c 249 576 c 2625 c
625c 252 56a
56a 7.483
2 2 28 11b 264 121b 2 57b
57b 7.550
Solution:
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 7 Using the Pythagorean Formula to Solve an Application
Slide 8.1 - 19
A rectangle has dimensions of 5 ft by 12 ft. Find the length
of its diagonal.
5 ft
12 ft
Solution:
2 2 25 12 c 225 144 c
2169 c
169c
13ftc
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 55
Use the distance formula.
Slide 8.1 - 20
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8.1 - 21
Use the distance formula.
The distance between the points and is
1 1,x y 2 2,x y
2 2
2 1 2 1 .d x x y y
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 8 Using the Distance Formula
Slide 8.1 - 22
Find the distance between and . 6,3 2, 4
2 22 6 4 3d
Solution:
224 7d
65d
16 49d
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 66
Find cube, fourth, and other roots.
Slide 8.1 - 23
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Finding the square root of a number is the inverse of squaring a
number. In a similar way, there are inverses to finding the cube
of a number or to finding the fourth or greater power of a
number.
The nth root of a is written
Find cube, fourth, and other roots.
.n a
Slide 8.1 - 24
n a
n a
Radical signIndex
Radicand
In , the number n is the index or order of the radical.
It can be helpful to complete and keep a list to refer to of third and fourth powers from 1-10.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Find each cube root. Solution:
EXAMPLE 9 Finding Cube Roots
Slide 8.1 - 25
3 64
3 27
3 512
4
3
8