chapter 8 section 1. objectives 1 copyright © 2012, 2008, 2004 pearson education, inc. evaluating...
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Chapter 8 Section 1
Objectives
1
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
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 theorem.
Use the distance formula.
Find cube, fourth, and other roots.
8.1
2
3
4
5
6
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Objective 1
Find square roots.
Slide 8.1-3
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
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
Square Root
A number b is a square root of a if b2 = a.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
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 positive or principal square root of a number is written with
the symbol .
0 0
a
Radical SignRadicand
The symbol is used for the negative square root of a number.
Slide 8.1-5
Find square roots. (cont’d)
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
The statement is incorrect. It says, in part, that a positive number equals a negative number.
9 3
Slide 8.1-6
Find square roots. (cont’d)
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Find all square roots of 64.
Solution:
Positive Square Root
Negative Square Root
64 8
64 8
Slide 8.1-7
EXAMPLE 1 Finding All Square Roots of a Number
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Find each square root.
Solution:
169
225
13
15
25
64
25
64 5
8
Slide 8.1-8
EXAMPLE 2 Finding Square Roots
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Find the square of each radical expression.
Solution:
17 2
17 17
31 2
31 31
22 3x 222 3x 22 3x
Slide 8.1-9
EXAMPLE 3 Squaring Radical Expressions
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Objective 2
Decide whether a given root is rational, irrational, or not a real number.
Slide 8.1-10
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Deciding whether a given root is rational, irrational, or not a real number.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
Slide 8.1-11
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
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.
Slide 8.1-12
EXAMPLE 4 Identifying Types of Square Roots
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Objective 3
Find decimal approximations for irrational square roots.
Slide 8.1-13
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Find decimal approximations for irrational square roots.
Even if a number is irrational, a decimal that approximates the number can be found using a calculator.
Slide 8.1-14
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
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
Slide 8.1-15
EXAMPLE 5 Approximating Irrational Square Roots
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Objective 4
Use the Pythagorean theorem.
Slide 8.1-16
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
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
Use the Pythagorean theorem.
2 2 2.a b c
Be careful not to make the common mistake thinking that
equals
2 2a b.a b
Slide 8.1-17
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
2 2 213 15a 2 169 225a
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:
Slide 8.1-18
EXAMPLE 6 Using the Pythagorean Theorem
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
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
Slide 8.1-19
EXAMPLE 7 Using the Pythagorean Theorem to Solve an Application
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Objective 5
Use the distance formula.
Slide 8.1-20
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Use the distance formula.
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
Slide 8.1-21
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Find the distance between and 6,3 2, 4 .
2 22 6 4 3d
Solution:
224 7d
65d
16 49d
Slide 8.1-22
EXAMPLE 8 Using the Distance Formula
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Objective 6
Find cube, fourth, and other roots.
Slide 8.1-23
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
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
In the number n is the index or order of the radical.,n a
n a
Radical sign
IndexRadicand
It can be helpful to complete and keep a list to refer to of third and fourth powers from 1-10.
Slide 8.1-24
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Find each cube root.
3 64
3 27
3 512
4
3
8
Slide 8.1-25
EXAMPLE 9 Finding Cube Roots
Solution:
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Find each root.
4 81
4 81
4 81
5 243
5 243
3
3
Not a real number.
3
3
Solution:
Slide 8.1-26
EXAMPLE 10 Finding Other Roots
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