Download UNIT 9 Rationals, Irrationals, and Radicals

UNIT 9
Rationals, Irrationals,
and Radicals
People of ancient times used a rope with knots for measuring right triangles.
332
UNIT 9
RATIONALS, IRRATIONALS, AND RADICALS
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Are rational numbers very levelheaded? Are irrational numbers hard
to reason with? Not really, but rational and irrational numbers have
things in common and things that make them different.
Big Ideas
►
A number is any entity that obeys the laws of arithmetic; all numbers obey the laws
of arithmetic. The laws of arithmetic can be used to simplify algebraic expressions.
►
If you can create a mathematical model for a situation, you can use the model to
solve other problems that you might not be able to solve otherwise. Algebraic
equations can capture key relationships among quantities in the world.
Unit Topics
►
Rational Numbers
►
Terminating and Repeating Decimals
►
Square Roots
►
Irrational Numbers
►
Estimating Square Roots
►
Radicals with Variables
►
Using Square Roots to Solve Equations
►
The Pythagorean Theorem
►
Higher Roots
RATIONALS, IRRATIONALS, AND RADICALS
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Rational Numbers
The rational numbers are a subset of the real
numbers. Examples of rational numbers are
3 , and 23.7__
1 , 5, −3, 0.25, 10 __
__
8.
4
2
DEFINITION
( )
a,
A rational number is any number that can be expressed as a ratio __
b
where a and b are integers and b ≠ 0.
The letter represents the set of rational numbers. Integers are rational
numbers because you can write them as a fraction (a ratio) with a
−6
denominator of 1. For example, −6 = ___
1 .
A proper fraction is a fraction where the numerator is less than the
denominator. An improper fraction is a fraction where the numerator is
greater than or equal to the denominator. Improper fractions have values
greater than or equal to 1 and can be written as mixed numbers. A mixed
number is a number consisting of both a whole number and a fraction or the
opposite of such a number.
Proper Fractions
7 __
3
1 __
1 __
−__
8, 3, 2, 4
Improper Fractions
26 __
16 ___
4 ___
12
−___
5 , 4, 8 , 5
NOTATION
The letter denotes the set of
rational numbers.
Mixed Numbers
7
2 , 1 __
1 __
−3 __
3 4, 5 8
Writing Rational Numbers
Example 1
A.
25
Write −___
8 as a mixed number.
Solution Write the improper fraction as a mixed number by dividing
the numerator by the denominator. Because a mixed number has a whole
25
number part, and whole numbers are not negative, consider −___
8 as the
25
opposite of ___
8.
(continued)
RATIONAL NUMBERS
335
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( )
25
1
__
− ___
8 = − 38
( )
1
= −3__
8
B.
Divide.
Simplify. ■
Write 4.25 as a percent, a mixed number, and an improper fraction.
REMEMBER
Solution
To write a decimal as a percent,
move the decimal point to the
right two places and add zeros
if necessary.
Write 4.25 as a mixed number.
Write 4.25 as a percent. 4.25 = 425%
25
4.25 = 4 ____
100
1
= 4 __
4
Write 0.25 as a fraction.
Simplify.
Write 4.25 as an improper fraction.
1
4.25 = 4 __
4
Write the decimal as a mixed number.
4·4+1
= ________
4
Multiply the whole number by the denominator and
add it to the numerator.
16 + 1
= ______
4
Multiply.
17
= ___
4
Add. ■
Comparing Rational Numbers
Two rational numbers are either equal or not equal to each other. If they are
not equal, then one of the numbers is greater than the other number.
COMPARISON PROPERTY OF RATIONAL NUMBERS
For positive integers a and c and nonzero integers b and d,
a > __
c if and only if ad > bc.
__
b
d
a < __
c if and only if ad < bc.
__
b
d
c if and only if ad = bc.
a = __
__
b
d
Example 2 Write <, =, or > to make a true statement.
A.
3
__
7
__
4
9
7
3
__
Compare the rational numbers __
4 and 9 . Use the comparison
c 7
a 3
and __ = __
.
property of rational numbers, with __ = __
b 4
d 9
Solution
ad = 3 · 9 = 27 and bc = 4 · 7 = 28
7
3 __
Since 27 < 28, __
4 < 9. ■
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UNIT 9
RATIONALS, IRRATIONALS, AND RADICALS
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B.
23
−___
9
18
−___
7
Solution Write each fraction as a mixed number.
( )
( )
23
18
_5_
_5_
___
_4_
_4_
− ___
9 = − 2 9 = −2 9 and − 7 = − 2 7 = −2 7
( )
( )
5
Compare _9_ and _47_. Multiply so you can use the comparison property of
rational numbers, ad = 5 · 7 = 35 and bc = 9 · 4 = 36. Since 35 < 36,
_5_
_4_
_5_
_4_
9 < 7 . That means −2 9 is closer to 0 on the number line than −2 7 and
23
5
18
___
−2 _9_ > −2 _74_. Therefore, −___
9 >−7. ■
THINK ABOUT IT
On a number line, a number to
the right of another number is
the greater of the two numbers.
Finding a Rational Number Between Two
Rational Numbers
The density property of rational numbers states that there are infinitely many
numbers between any two rational numbers.
Example 3
3
3
Find a rational number between _4_ and _5_.
Solution One solution is to find the number halfway between the two
numbers. Find the average of the two numbers by finding their sum and
dividing by 2.
Step 1 Add the numbers.
Step 2 Divide the sum by 2.
_3_
27
___
_3_
_3_ _5_
_3_ _4_
4+5=4·5+5·4
27 _2_
___
20 ÷ 2 = 20 ÷ 1
TIP
= 20 + 20
= 20 · 2
Check the answer to Example 3
by writing each fraction as
a decimal.
27
= ___
20
27
= ___
40
3 = 0.6
3 = 0.75, ___
27 = 0.675, and __
__
15
___
12
___
27 _1_
___
4
40
5
27
_3_
_3_
The rational number ___
40 is between 4 and 5 . ■
Proving that the Rational Numbers Are Closed
Under a Given Operation
Recall that a set of numbers is closed under an operation if the result of the
operation with two numbers in the set is also a member of that set. The rational
numbers are closed under addition, subtraction, multiplication, and division.
Example 4 Prove that the rational numbers are closed under multiplication.
Solution For all integers a and c and nonzero integers b and d,
c
ac
ac
_a_
· __ = ___ . The number ___ is a rational number because the set of
b d
bd
bd
integers is closed under multiplication. The denominator cannot be zero because neither b nor d is zero. ■
RATIONAL NUMBERS
337
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Problem Set
Write each number as a mixed number, proper or improper fraction,
decimal, and percent, if possible.
1.
7
___
2.
2.8
20
3.
0.3125
5.
1
11__
8
4.
15
___
6.
13
___
2
5
Write <, =, or > to make a true statement.
5
−__
6
7.
8.
8
___
11
___
13
16
1
12 __
3
9.
6
−__
7
37
___
3
36
___
29
___
15
8
11.
5
7 __
8
31
___
12.
−0.71
10.
13.
14.
4
25
−___
19
4
−1___
17
13
___
169
____
14
182
7
−__
9
Arrange each list of numbers in increasing order.
7 ___
8 ___
11
__
15.
16.
8 , 10 , 13
18.
7
5
3 __
__
−___
13 , − 5 , − 8
21.
9 ___
13
2 __
−3__
3, −2, − 3
19.
27
64 ___
2.58, ___
25 , 10
22.
20.
130 ____
140 ___
45 ___
52
−____
63 , − 46 , − 6 , − 9
7
5 __
3 __
1 __
__
3, 5, 7, 9
17.
69
64 ___
___
9 , 10 , 6.4
99 ___
79
53 ____
___
68 , 104 , 85
Find a rational number between the two given numbers.
6, 9
25.
47
2 ___
9 __
3, 8
17
14 ___
−___
31 , −33
26.
39
4 ___
−3__
5 , − 11
5 __
5
__
23.
24.
27.
2
1, __
3
Solve.
Prove that the rational numbers are closed under:
28.
Challenge
7
__
2
__
8
3
A. addition
A. Find a rational number between − and − .
B. subtraction
B. Determine if your answer is less than −0.7. If
C. division
Challenge
*29.
*30.
A. Find a rational number between
10
___
5 and 13.
3
__
7
not, find a rational number between −__
8 and
2
−__
3 that is less than −0.7.
B. Determine if your answer is greater than 0.7.
3
If not, find a rational number between __
5
10
and ___
that
is
greater
than
0.7.
13
338
UNIT 9
RATIONALS, IRRATIONALS, AND RADICALS
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Terminating and Repeating Decimals
There are different types of decimals.
DEFINITIONS
TIP
Terminating decimals are decimals that have a finite number of nonzero
digits. Nonterminating decimals are decimals that do not terminate,
or end.
The word terminate means stop.
Terminating decimals stop;
nonterminating decimals do
not stop.
There are two types of nonterminating decimals: repeating and nonrepeating
decimals. Repeating decimals have a repeating pattern of digits, while nonrepeating decimals do not. Place a bar over the block of digits that repeat in
a repeating decimal.
Terminating
Decimals
_
0.25, 6, −1.5836
NOTATION
Nonterminating Decimals
Repeating
0.4___
= 0.44444. . .
3.256_= 3.256256. . .
20.763 = 20.76333. . .
A bar placed over digits
in a decimal shows the
digits repeat.
Nonrepeating
0.356987412569112. . .
3.1415926535. . .
Converting Fractions to Decimals
Example 1 Express each fraction as a decimal. Determine if the decimal
repeats or terminates.
A.
7
___
25
Solution Divide 7 by 25: 7 ÷ 25 = 0.28. The decimal is a terminating
decimal. ■
B.
TIP
On a calculator, the last digit
displayed may be rounded up,
even when the digits continue
to repeat.
7
__
9
_
Solution Divide 7 by 9: 7 ÷ 9 = 0.777777. . . = 0.7. The decimal is a
repeating decimal. ■
C.
REMEMBER
5
___
12
_
Solution Divide 5 by 12: 5 ÷ 12 = 0.41666666. . . = 0.416. The decimal is
a repeating decimal. ■
Place the bar over the repeating
part of the decimal only.
TERMINATING AND REPEATING DECIMALS
339
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Expressing Terminating and Repeating Decimals
as Fractions
A number is rational if and only if you can write it as a terminating or repeating decimal. Both terminating and repeating decimals can be written as the
quotient of two integers.
Example 2
A.
Write each decimal as a quotient of two integers.
4.25
Solution Since the last digit in the decimal part is in the hundredths place,
write 0.25 as a fraction with a demonimator of 100.
17
25
1
1 ___
__
__
4.25 = 4 + 0.25 = 4 + ____
100 = 4 + 4 = 4 4 = 4 ■
__
B.
4.25
Solution Write the number as an equation. Multiply each side of the equation by the power of 10 that has as many zeros as there are digits in the
repeating block.
__
x = 4.25
__
100x = 425.25
Subtract the first equation from the second. This will eliminate the repeating
part. Then isolate the variable and simplify the fraction if possible.
__
100x = 425.25
−
x=
__
4.25
99x = 421
TIP
99x
____
When multiplying both sides
of the equation by a power
of 10, it helps to write the
repeating decimal without the
bar and display the digits in
the repeating block a couple
of times.
421
____
99 = 99
421
x = ____
99
__
Check 421 ÷ 99 = 4.25 ■
___
C.
2.02342
Solution
Multiply each side of the equation by 1000.
___
x = 2.02342
___
1000x = 2023.42342
Subtract the first equation from the second.
___
1000x = 2023.42342
−
x=
___
2.02342
999x = 2021.4
Multiply each side by 10 to eliminate the decimal. Then isolate the variable
and simplify.
9990x = 20,214
20,214
9990x ______
______
9990 = 9990
1123
x = _____
555 ■
340
UNIT 9
RATIONALS, IRRATIONALS, AND RADICALS
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Finding a Rational Number between a Fraction
and a Repeating Decimal
__
8
Find a rational number between ___
81.
and
0.
11
Example 3
Solution
Step 1 Write the repeating decimal as a fraction.
__
100x = 81.81
__
−
x = 0.81
99x = 81
9
81 ___
x = ___
99 = 11
Step 2 Find the average of the two fractions.
9
17
8
___
___
Add the numbers. ___
11 + 11 = 11
17
17 __
17
1 ___
___
Divide the sum by 2. ___
11 ÷ 2 = 11 · 2 = 22
__
17
8
___
81.
The number ___
is
between
and
0.
22
11
17
8
___
Step 3 Check. Write ___
11 and 22 as decimals.
__
__
17
___
27
and
27
=
0.7
=
0.77
11
22
8
___
__
__
__
0.727 < 0.7727 < 0.81 ■
Application: Proof with Repeating Decimals
_
Prove that 0.9 = 1.
Example 4
_
Solution Write 0.9 as a fraction.
_
Let x = 0.9.
_
10x = 9.9
_
−
x = 0.9
9x = 9
9
x = __
9=1
_
_
Since x = 0.9 from the first assumption, and x = 1 from the algebra, 0.9 = 1
by the substitution property of equality. ■
TERMINATING AND REPEATING DECIMALS
341
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Problem Set
Express each fraction as a decimal. Determine if the decimal repeats or
terminates.
1.
3
__
2.
2
___
3.
1
__
4
13
9
4.
9
__
5.
4
___
6.
7
__
5
15
7.
23
___
8.
10
___
6
27
2
Write each decimal as a quotient of two integers.
7.2
12.
10.
5.13
11.
0.125
9.
270.35
15.
2.54
13.
0.074
16.
3.2
14.
1.72
17.
4.07
26.
_
11
−3.5 and −___
3
27.
__
7
−___
72
and
−0.22
24
30.
1999.9 = 2000
_
___
_
__
_
Find a rational number between the two given numbers.
__
_
5
__
3 and 0.45
18. ___
22. 0.136 and
6
10
_
__
19.
3 and 1.8
__
23.
11 and −1.318
−___
6
20.
__
10
0.681 and ___
11
24.
_
1
0.083 and ___
13
21.
_
1
0.16 and __
3
25.
32 and 2.4
___
29.
19.9 = 20
2
_
15
Prove each statement.
_
_
2.9 = 3
28.
_
Solve and show your work.
Challenge
*31.
A. Write
13
___
1
__
50 and 4 as decimals.
B. Find a repeating decimal between these two
numbers.
C. Write your answer for Part B as a fraction.
342
UNIT 9
*32.
Challenge
A. Write
7
___
8
___
11 and 11 as decimals.
B. Find a repeating decimal between these two
numbers.
C. Write your answer for Part B as a fraction.
RATIONALS, IRRATIONALS, AND RADICALS
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Square Roots
The result of multiplying a number by itself is the
square of the number.
DEFINITION
A square root is the factor of a number that when multiplied by itself
results in the number.
Every positive real number has two square roots, one positive and one negative. If x2 = a, then x · x = a and (−x) · (−x) = a. For example, 9 is the
square of 3 because 3 · 3 = 9. Notice also that (−3) · (−3) = 9. Since both 32
and (−3)2 equal 9, both 3 and
___ −3 are square roots of 9.
is used, only the positive square root is being
When the radical
sign
√
__
asked for. So, √9 = 3. The positive square root is also called the principal, or
nonnegative, square root.
NOTATION
___
The radical sign √ indicates
the principal square root.
Finding Square Roots
Example 1
A.
THINK ABOUT IT___
Evaluate.
The square root of 100.
Solution The square roots of 100 are −10 and 10 because (−10)2 = 100
and 102 = 100. ■
___
B.
√ 49
√ can also be
The radical sign
2 ___
written as √
to indicate the
second, or square, root. Most
books omit the 2.
Solution The square roots of 49 are 7 and___−7, but since the radical sign is
used, give only the principal square root: √49 = 7. ■
__
C.
√__94
__
√
4 __
2
Solution Use the principal square root. Since 3 · 3 = 9 , __
9 = 3. ■
2 __
2
__
____
D.
4
__
−√1.44
Solution Find the opposite of the principal square____
root. The principal
square root is 1.2. The opposite of 1.2 is −1.2. So, −√1.44 = −1.2. ■
SQUARE ROOTS
343
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Like Radicals
__
In the square root expression 3√7 , 3 is the coefficient and 7 is the radicand.
When two or more square root expressions have the same radicand, the
expressions are like radicals and can be combined by adding or subtracting
the coefficients.
THINK ABOUT IT
Like Terms
Like Radicals
__
__
__
3√___
7 + 5√___
7 = 8√7___
3x + 5x = 8x
3y − y = 2y 3√14 − √14 = 2√14
Combining like radicals is similar
to combining like terms.
Example 2 Simplify.
__
__
__
4√2 + 9√2 + 9√3
A.
Solution
__
__
__
__
Identify the like radicals: 4√2 and 9√2 . Add their coefficients.
__
4√2 + 9√2 + 9√3
__
__
= 13√2 + 9√3
__
__
The sum is 13√2 + 9√3 . These two expressions cannot be combined because
their radicands are different. ■
__
__
−9√5 − 2√5
B.
Solution
The radicals are like radicals. Subtract the coefficients.
__
__
−9√5 − 2√5
__
= −11√5
■
Computing Products of Square Roots
The product of two__square
__ root
___expressions is the square root of the product
of the radicands: √a · √b = √ab .
__
__
Example 3 Multiply √5 · √2 .
Solution
__
__
____
___
Multiply the radicands: √5 · √2 = √5 · 2 = √10 . ■
Application: Verifying and Justifying Facts About
Square Roots
Example 4
__
__
_____
Verify that √a + √b ≠ √a + b .
A.
Solution
___
___
Substitute nonzero numbers for a and b. Let a = 16 and b = 25.
_______
√ 16 + √ 25
√16 + 25
4+5
√ 41
9
___
__
≈ 6.4
__
_____
9 ≠ 6.4, so √a + √b ≠ √a + b ■
344
UNIT 9
RATIONALS, IRRATIONALS, AND RADICALS
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B.
Explain why a negative number cannot have a real square root.
Solution A square root of a number is a number that, when multiplied by
itself, equals the original number. Any number times itself cannot be a negative number because the product of two positive numbers is positive and the
product of two negative numbers is positive. ■
Problem Set
Evaluate.
___
___
1.
√ 81
√ 36____
5.
√169
4.
−√0.04
7.
1600
8.
0.25
3.
__
2.
−√9
____
1
___
121
____
Find the square roots of each number.
6.
144
____
49
Simplify.
__
√ 7 + 7√ 7
10.
9√2 − 3√2
__
__
__
__
___
√3 · √5
12.
√ 2 · √ 1.1
15.
38
√91 __+ √___
91
__
___
71
___
−4√2 − 4√2
2√7 + 3√6 + 4√7 + 2√6
21.
√ 5 · √ 3 · √ 6 · √ 10
√__7 · √__6 + √__14 · ___
√3
22.
8√82 + 2√82
23.
( 3√2 + 2√2 − 4√2 ) · √2
24.
√ 1.5 · √ 150
__
___
___
5
__
2
__
1
___
10
___
√ 5 · √ 2 + √ 3 · √ 10
___
___
____
__
___
___
−3√26 + 3√13 + 10√26 − 3√13
____
__
__
__
__
__
__
17.
19.
___
__
___
16.
___
15√17 + 25√34 − 10√17
20.
__
14.
___
18.
__
11.
13.
___
__
9.
___
√ 2 · √ 0.33 + √ 0.06 · √ 11
25.
___
___
__
__
___
____
___
___
__
__
___
1
√ 30 · √ 42 · √___
35
1
___
1
___
Solve.
26.
64 2
A square has an area of ____
169 in . Find the side
length.
*29.
Challenge Verify that the set of irrational
numbers is not closed under multiplication.
*30.
Challenge
___
27.
28.
A rectangle ___
has a length of √43 meters and
a width of √21 meters. Find the area of the
rectangle.
A circle has an area of 256π square units. Find
the radius.
__
__
A. Prove that √ 8 = 2√ 2 .
__
__
B. Find √ 3 · √ 6 and simplify.
SQUARE ROOTS
345
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Irrational Numbers
Real numbers that are not rational numbers are
irrational numbers.
NOTATION
The letter denotes the set of
irrational numbers.
Unlike a rational number, an irrational number is a real number that cannot
a
be written in the form __, for any integers a and b. The set of rational numb
bers and the set of irrational numbers make up the set of real numbers.
⺢: Real Numbers
⺡: Rational Numbers
⺪: Integers
⺧: Whole Numbers
⺙: Irrational Numbers
⺞: Natural Numbers
Determining if a Number Is Rational or Irrational
REMEMBER
π = 3.14159. . .
Any decimal that is nonterminating and nonrepeating is an irrational number.
The number π is an example of an irrational number. There is no way to
convert the decimal into a fraction of integers because there is no repeating
block of digits.
A perfect square is a rational number whose square root is also rational.
4. The square root of any numExamples of perfect squares are 9, 25, and __
9
__
___
ber that is not a perfect square is an irrational number, such as √2 and √14 .
THINK ABOUT IT
A decimal cannot accurately
represent an irrational number,
but a decimal
can approximate
__
its value. √2 is an exact
value;
__
1.414 approximates √2 .
Rational Numbers
__
___
___
√ 9 = 3, √ 25 = 5, √ 36 = 6
Example 1
___
A.
√ 90
Solution
____
B.
√ 256
Irrational Numbers
__
2=
√___
1.414213562. . .
√ 14 = 3.741657387. . .
Determine if each number is rational or irrational.
___
Because 90 is not a perfect square, √90 is irrational. ■
____
Solution 256 is a perfect square because 16 · 16 = 256. √256 is
rational. ■
346
UNIT 9
RATIONALS, IRRATIONALS, AND RADICALS
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VHS_ALG_S1.indb 346
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____
−√121
C.
Solution The
____square root of 121 is 11. The opposite of 11 is −11, which is
rational. −√121 is rational. ■
__
√__51
D.
Solution Because no number times itself
__ equals 5, there is no number that
1
1
__
can be multiplied by itself to equal 5 . __
5 is irrational. ■
√
___
64
√___
49
E.
___
______
64 __
8
=
1.
142857,
Solution Both 64 and 49 are perfect squares. Since ___
=
49 7
___
64
___
49 is rational. ■
√
√
Simplifying Radicals
___
__
__
To simplify radicals, use √ab = √a · √b . When a radical expression contains
no radicands with factors that are perfect squares other than 1, the expression
is in simplified radical form.
Example 2
Simplify.
___
√ 40
A.
Solution Choose two factors of 40 so that one of them is a perfect square.
___
_____
√ 40 = √ 4 · 10
__
___
= √4 · √10
___
= 2√10
Ten is not a perfect square
___ and there are no
___factors of 10 that are perfect
squares other than 1, so √40 simplifies to 2√10 . ■
____
B.
√ 108
Solution Choose two factors of 108 so that one of them is a perfect
square.
____
_____
√ 108 = √ 36 · 3
___
__
= √36 · √3
TIP
Make a list of the first 15
perfect squares to refer to while
simplifying radicals.
__
____
= 6√ 3
__
√ 108 simplifies to 6√ 3
It is possible to choose a perfect square factor that is not the greatest perfect
square factor. Suppose you chose 4 and 27.
____
_____
√ 108 = √ 4 · 27
__
___
= √4 · √27
___
= 2 · √27
__
__
= 2 · √9 · √3
__
__
= 2 · 3 · √3 = 6√3 ■
IRRATIONAL NUMBERS
347
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Verifying the Closure Properties for Irrational Numbers
The irrational numbers are closed under addition and subtraction.
Example 3 Determine if the set of irrational numbers is closed under
multiplication.
Solution The product of two irrational numbers is not always an irrational
number. Here is an example.
__
__
__
√3 · √3 = √9 = 3
The set of irrational numbers is not closed under multiplication. ■
Application: Proving That a Square Root Is Irrational
The square of an even number is always an even number and the square of
an odd number is always an odd number. Use these facts in the following
example.
__
Example 4 Prove that √2 is irrational.
THINK ABOUT IT
a is not a simplified fraction,
If __
b
then common factors can be
divided out until it is, resulting in
an equivalent fraction.
__
Assume that √2 is rational. Then it would be possible to represent
a
the value as a simplified fraction __, where a and b are integers and b is not
b
zero. Because the fraction is simplified, a and b have no common factors
other than 1.
Solution
__
√2 =
__
a
__
b
a
( √2 )2 = __
2
(b)
a2
2 = __2
b
2b2 = a2
Square both sides of the equation.
__
__
__
√ 2 · √ 2 = √ 4 = 2 and
a2
= __2
b b b
a __
a
__
·
Means-Extremes Product Property
Since a number with a factor of 2 is an even number, 2b2 is an even number.
That means a2 is an even number, and since only an even number squared can
result in an even number, a must also be even.
Since a is an even number, it can be written as a product with a factor of 2.
Let a = 2c, where c is an integer.
2b2 = a2
Last line from above
2b2 = (2c)2
Substitution Property of Equality
2b = 4c
2c · 2c = 4c2
2
2
b2 = 2c2
TIP
When an assumption leads to a
contradiction (two statements
with opposite ideas), the
assumption is false.
348
Divide each side by 2.
2
UNIT 9
Now b is shown to be an even number because it is equal to a product with
a factor of 2. Since b2 is even, b is even.
Both a and b have been shown to be even, which means they both have a
common factor of 2. However, it was stated in the beginning of the proof that
a and b have no common factors
other than 1. This is a contradiction. There__
fore, the assumption that √__2 is rational must be incorrect. A number is either
rational or irrational, so √2 is irrational. ■
RATIONALS, IRRATIONALS, AND RADICALS
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Problem Set
Determine if each number is rational or irrational. Explain.
___
___
1.
√ 75
4.
−√49
2.
√ 100
5.
√ 144
3.
√7
6.
π
15.
√9
16.
√____
81
____
__
3
__
___
√____
9
4
__
√ 100
17.
12.
√ 125
13.
14.
11.
8.
√25
__
√ 72
__
10.
−√170
____
Simplify.
9.
____
7.
___
16
___
____
21.
___
−√288
__
36
___
22.
1
__
√____
4
√ 196
23.
√ 169
18.
−√162
24.
√ 27
−√12
19.
√ 300
25.
5√8
√ 45
20.
√ 121
____
____
___
___
____
___
__
____
Solve.
26.
Show that the set of irrational numbers is not
closed under division. That is, show that there
are two irrational numbers with a quotient that is
not an irrational number.
*27.
Challenge The unit square diagrams show that
every perfect square can be written as a sum of
the form 1 + 3 + 5 + 7 + . . .
1
4=1+3
*28.
Challenge Write an example to illustrate the
process of finding the greatest perfect square
divisor of x. Choose a value a and test to see if
x
x is an integer. If __
__
is an integer, a is the
a2
a2
greatest perfect square divisor of x. Begin
__ with
the least value of a that is greater than √x and
work backwards.
9=1+3+5
A. Complete the list below:
1=1
4=1+3
9=1+3+5
16 =
25 =
36 =
B. Predict the number of addends in the expan-
sion for 144 and 1024.
IRRATIONAL NUMBERS
349
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Estimating Square Roots
REMEMBER
Irrational numbers
are nonterminating,
nonrepeating decimals.
Because the square root of a number that is not a
perfect square is an irrational number, its decimal
value can only be estimated.
PROPERTY
For nonnegative values of m, n, and p,
__
__
__
if m < n < p, then √ m < √n < √ p .
If a number is between two other numbers, then its square root is
between
roots of the other numbers. To illustrate, 3 < 4 < 5 and
__
__ the square
__
√3 < √4 < √5 .
3⬇1.73
1
2
5⬇2.24
3
4
5
4=2
Determining the Location of the Square Root of a
Non-Perfect Square
Example 1
root lies.
Determine between which two consecutive integers each square
___
A.
√ 52
Solution Think of the perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81,
100, ... Choose the nearest perfect
___ squares less
___ than and
___greater than 52.
They are 49 and 64. Because √49 = 7 and √64 = 8, √52 lies between 7
and 8. ■
____
B.
√ 125
2
Solution Since
____ 125 lies between the perfect squares 121 (11 ) and
2
144 (12 ), √125 lies between 11 and 12. ■
350
UNIT 9
RATIONALS, IRRATIONALS, AND RADICALS
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Estimating Square Roots of Non-Perfect Squares
To estimate the square root of a number, first determine between which two
consecutive integers the square root lies. Adjust the estimate based on which
perfect square the number is closer to and by how much.
___
Example 2
Estimate √23 to the nearest tenth.
___
Solution Twenty-three lies between 16 and 25, so √23 lies between
___ 4 and
5. Since 23 is about three-quarters of the way between 16 and 25, √23 should
be about three-quarters of the way between 4 and 5, or at about 4.75. Test
values around 4.75.
4.72 = 22.09 and 4.82 = 23.04
___
So, √23 ≈ 4.8. ■
Using the Babylonian Method to Estimate a Square Root
Another method to use when estimating the square root of a number is the
Babylonian method.
USING THE BABYLONIAN METHOD TO FIND THE
SQUARE ROOT OF x
THINK ABOUT IT
Step 1 Start with any guess r1 of the square root.
x to compute a new guess r .
Step 2 Find the average of r1 and __
2
r1
The first guess is r1, the second
guess is r2, and the nth guess
is rn .
x are as
Step 3 Repeat Step 2 using r2. Continue this process until rn and __
rn
close as desired.
__
Example 3 Use the Babylonian method to estimate √7 to the nearest
hundredth.
Solution
Step 1 Guess 2.5 because 2 · 2 = 4 and 3 · 3 = 9.
x
7
r1 + __
2.5 + ___
r1 ________
2.5 + 2.8 ___
2.5 ________
5.3
_______
Step 2
=
=
= 2 = 2.65
2
2
2
x
7
r2 + __
2.65 + ____
r
2.65 + 2.64 ____
5.29
2.65 __________
__________
______2
=
Step 3
≈
≈ 2 ≈ 2.645 ≈ 2.65
2
2
2
The estimates after Step 2 and Step 3 are so close that__ the hundredths place
doesn’t change, so this is a good stopping point. So, √7 ≈ 2.65. ■
ESTIMATING SQUARE ROOTS
351
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Application: Geometry
Example 4 Find the perimeter of a square whose area is 450 square meters.
Round your answer to the nearest tenth of a meter.
Solution The side length of a square is the square root of the
area. Perim____
eter is four times the length of one side, so the expression 4√450 represents
the perimeter of the square.
____
4√450 ≈ 84.9
The perimeter is about 84.9 meters.
2
= 450.500625 ≈ 450 ■
( 84.9
4 )
____
Check
Problem Set
Determine between which two consecutive integers each square
root lies.
___
____
___
1.
√ 19
4.
√ 240
7.
√ 40
2.
√ 28
5.
√ 70
8.
√ 61
3.
√ 12
6.
√ 92
9.
√ 136
___
___
___
___
___
____
Estimate the given square root to the nearest tenth.
___
___
__
10.
√ 17
14.
√5
18.
√ 23
11.
√ 78
15.
√ 150
19.
√3
12.
√ 45
16.
√ 20
20.
√ 60
13.
√ 85
17.
√ 55
___
____
___
__
___
___
___
___
Use the Babylonian method to estimate the given square root to the
nearest tenth.
___
____
√ 30
21.
√ 200
____
___
23.
√ 165
Complete____
the missing information in the problem
to find √300 using the Babylonian method.
26.
The area of a square is 45 square units. Estimate
the side length.
First Guess: 17
27.
Five times the square of a number is 165.
Estimate the number.
28.
The quotient of the square of a number and 2 is
0.6. Estimate the number.
29.
The area of a circle is 20π m2. Find the
approximate radius of the circle.
22.
24.
√ 99
Solve. When estimating, round to the nearest tenth.
25.
300
17 + ____
+ 300
________
17
________
2
2
________ =
=
589
____
17
____
≈
.3235;
300.1036523 + 300
_________________
+ _______
17.3235
17.3235 = _________________
____________
=
2
2
_______
17.3235 ≈ 17.32 ≈ 17.3
_______
2
352
UNIT 9
*30.
Challenge Square a number, divide the result
by 10 and multiply that result by 2 to obtain 57.8.
What is the number?
RATIONALS, IRRATIONALS, AND RADICALS
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Radicals with Variables
Radical expressions may contain variables.
The square root of a nonnegative
number
is squared
__
__ that____
__ is the original
3 = √9 = 3.
number. For example, √32 = 3 because √32 = √3 ·__
Suppose the base of the power is__a variable: √a2 . Since the radical sign
indicates the principal square root, √a2 = a.
Evaluating a Variable Square-Root Expression
When evaluating a variable square-root expression, treat the radical sign as
a grouping symbol.
Example 1 Evaluate. If necessary, round your answer to the nearest
hundredth.
______
√ 11 − a , when a = 2 and when a = −3
A.
Solution
______
__
√ 11 − 2 = √ 9 = 3
_________
___
= √14 ≈ 3.74 ■
√11 − (−3) ___
______
10 − √2d + 2√d + 15 , when d = 15
B.
Solution
_____
_______
10 − √2 · 15 + 2√15 + 15
___
___
10 − √30 + 2√30
Simplify under the radical signs.
___
10 + √30
Add like radicals.
≈ 15.48
________________
Find the sum. Then round. ■
4√(b + c) + (b − 5) , when b = 7 and c = 6
C.
2
2
Solution
________________
4√(7 + 6)2 + (7 − 5)2
_______
4√132 + 22
Simplify inside the parentheses.
4√169 + 4
Evaluate the powers.
_______
____
4√173
≈ 52.61
Add.
TIP
Rounding should always be the
last step in evaluating a squareroot expression.
Find the product. Then round. ■
RADICALS WITH VARIABLES
353
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Simplifying Square Roots of Powers with Variable Bases
Example 2 Simplify each expression. Assume all variables are nonnegative.
____
√x6y2
A.
____
______
√x6y2 = √(x3)2y2 = x3y
____
Solution
√ 16a5
B.
____
_____
■
_______
__
Solution √16a5 = √16a4a = √42(a2)2a = 4a2√a ■
_______
TIP
2√150p18q3
C.
The square root of a power with
an exponent that is an even
number is a perfect square.
Solution
_______
___________
2√150p18q3 = 2√52 · 6(p9)2q2q
___
= 2 · 5p9q√6q
___
= 10p9q√6q ■
Simplifying Radical Expressions with Quotients
Just as the product of two square roots is the square root of the product, a
quotient of two square roots is the square root of the quotient.
PROPERTY
For nonnegative values a and b, where b ≠ 0,
__
__
√b
√__
a = __
a
___
√b
Example 3 Simplify each expression.
__
√__9x
THINK ABOUT IT
A.
x is simplified
The expression ___
3
__
because the radicand √ x does
not contain a quotient.
Solution
simplify.
__
√
__
Rewrite the expression as a quotient of two radicals. Then
__
x
√__
___
__
√x
___
√ 9 = √9 = 3
x
__
■
___
B.
√ 72
____
__
√2
Solution Rewrite the expression as the square root of the quotient of
the radicands.
___
___
___
72
=
= ___
√ 36 = 6 ■
2
√2
√ 72
____
__
354
UNIT 9
√
RATIONALS, IRRATIONALS, AND RADICALS
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Rationalizing the Denominator
A radical expression is not considered simplified if a radical appears in a
denominator. To clear a radical from a denominator, you can rationalize
the denominator. To rationalize a denominator, multiply the numerator and
denominator of the expression by the radical in the denominator of the
expression.
Example 4
__
THINK ABOUT IT
__
Simplify each expression.
√__7x
A.
7 is
__
The expression ___
√7
equivalent to 1, so multiplying
an expression by it does
not change the value of
the expression.
√
Solution
__
__
x
√__
___
√ 7 = √7
x
__
__
=
B.
√7
__ __
__
7
x ___
√__
√__
___
·
√7
=
√x
√ 7___
_____
√ 49
___
√ 7x
____
= 7
■
−4y
____
__
TIP
√8
Solution
−4y
−4y
−4y
−2y
____
__ = _____
__ __ = ____
__ = ____
__
2√2
√4 √2
√2
√8
__
__
__
2y
−2
−2y
√
2
√
____
___
______
= __ · __ = 2 = −√2 y ■
√2 √2
Simplifying the radical in the
denominator first will minimize
reducing later in the problem.
Application: Analyzing a Bogus Proof
Example 5
32
32
__
The statement 1 − __
=
2
−
2
2 is true.
(
) (
)
Find the error in the following proof that 1 ≠ 2.
Step 1
2
2
Step 2
Step 3
2
( 1 − __32 ) = ( 2 − __32 )
_______
_______
3
__
√( 1 − 2 ) = √( 2 − __23 )
Given
2
Take the square root of each side.
3
3
__
1 − __
2=2−2
Step 4
1=2χ
3
Add __
2 to each side and simplify.
Therefore, 1 ≠ 2.
3
3
__
Solution Simplify 1 − __
2 and 2 − 2.
3
3 __
1
1
__
__
1 − __
2 = −2 and 2 − 2 = 2
This means that in Step 2, when the square root of each side is taken, the
square root of a negative number was taken on the left side. This step is
not allowed. ■
RADICALS WITH VARIABLES
355
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Problem Set
Simplify. Assume all variables are nonnegative.
__
____
√x4y10
1.
8.
√4
15.
√___11x
9.
√ 36
____
__
16.
2x
____
___
17.
3x
___
−____
√ 27
___
____
√ 25x
9
2.
3√90a12b15
√ 128
_____
__
11.
√25
18.
x
√___
13
12.
√ 700
_____
__
19.
4x
____
−_____
√ 108
√x y
7 8
√ 72x
5.
√7
___
13.
√121
20.
x
√___
10
14.
√ 80
____
__
21.
5a
____
___
√ 64x
13
a
____
___
________
4√50x y z
14 11 2
7.
___
x
___
____
_____
6.
2
√
___
____
____
3
√ 18
10.
____
4.
√4
____
_______
3.
___
x
__
√5
√ 84
Evaluate each expression for the given values. Provide answers in
simplified radical form and approximate irrational answers to the
nearest hundredth.
______
_____
√x + 7
22.
26.
A. x = 10
B. x = −2
_____
√ 2x − 1
A. x = 1
B. x = 3
_______________
___
23.
15 + 2√a + 8 − √3a , when a = 4
27.
3√(ab)2 + (2a + b)3 , when a = 1 and b = 4
24.
2√(x + y) − (x − y)2 , when x = 8 and y = 5
28.
√2x + 3 − 10√x + 1 − 5, when x = −1
25.
20 + √4 − x − 3√x , when x = 2
_______________
_____
______
__
_____
Solve.
1 2 = __
1−2
Challenge The statement 1 + __
2
2
is true. Find and explain the error in the
following proof.
(
*29.
1 2 = __
1−2
1 + __
2
2
) (
2
( ) ( )
_______
_______
1
__
√( 1 + 2 ) = √( __21 − 2 )
2
2
1 = __
1
1 + __
2 2−2
2
)
*30.
Challenge The statement (1 − 4)2 = (1 + 2)2 is
true. Find and explain the error in the following
proof.
(1 − 4)2 = (1 + 2)2
_______
_______
√(1 − 4)2 = √(1 + 2)2
1−4=1+2
−4 = 2 χ
1 = −2 χ
356
UNIT 9
RATIONALS, IRRATIONALS, AND RADICALS
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Using Square Roots
to Solve Equations
An equation can have zero, one, two, or even an
infinite number of solutions.
When a solution of an equation is substituted for the variable, the statement
is a true statement.
Using Square Roots to Solve Equations
Consider the equation x2 = 9. This equation has two solutions because both
3 and −3, which are the square roots of 9, make the equation true. To solve
an equation in which the variable
is squared, take the square root of each side
__
of the equation. Because √x2 = |x|,
__ and both the positive and negative square
roots are solutions, write x = ±√9 .
PROPERTY
__
NOTATION
The symbol ± is read as “plus
or
__
minus.” The expression ±√ 9 is
an__abbreviated
way of writing
__
“√ 9 and −√9 .”
For nonnegative values of a, if x2 = a, then x = ±√ a .
Example 1 Solve the equation. If necessary, round your answer to the
nearest tenth.
A.
n2 = 25
Solution
n2 = 25
___
n = ±√25
Take the square root of each side.
n = ±5
Simplify √25 .
___
The solutions are 5 and −5. ■
5t2 = 31
B.
Solution Isolate the variable and take the square root of each side.
5t2 = 31
31
t2 = ___
5
Divide each side by 5.
___
√___
31
t = ± ___
5
Take the square root of each side.
t = ±√6.2
Simplify the radicand.
t ≈ ±2.5
Estimate √6.2 with a calculator.
___
___
___
The solutions are exactly √6.2 and −√6.2 or about 2.5 and −2.5. ■
(continued)
USING SQUARE ROOTS TO SOLVE EQUATIONS
357
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9x2 − 64 = 0
C.
Solution
Use inverse operations to isolate the variable.
9x − 64 = 0
2
9x2 = 64
64
x2 = ___
9
Add 64 to each side.
Divide each side by 9.
___
√___
64
x = ± ___
9
√ 64
__
x = ±____
√9
8
x = ±__
3
Take the square root of each side.
Write the square root as a quotient.
___
__
Simplify √64 and √9 .
8
8
__
The solutions are __
3 and −3. ■
Applications: Geometry and Physics
Example 2 The area of a circle is 40 square centimeters. Estimate the radius
to the nearest tenth of a centimeter.
Solve the area formula A = πr 2 for r.
Solution
40 = πr 2
Substitute 40 for A.
40
___
Divide each side by π.
π = r
2
___
√
40
± ___
π = r
Take the square root of each side.
±3.6 ≈ r
Use a calculator to estimate.
Because length cannot be negative, disregard the negative answer. The radius
is about 3.6 centimeters. ■
Example 3 Solve −16t2 + 80 = 0 to estimate the number of seconds t it
takes for an object dropped from 80 feet above the ground to hit the ground.
Solution
−16t2 + 80 = 0
−16t2 = −80
t2 = 5
__
t = ±√5 ≈ ±2.236
Subtract 80 from each side.
Divide each side by −16.
Take the square root of each side.
Since time cannot be negative, disregard the negative answer. The time is
about 2.24 seconds. ■
358
UNIT 9
RATIONALS, IRRATIONALS, AND RADICALS
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Problem Set
Solve each equation. If necessary, round your answers to the
nearest tenth.
1.
x2 = 64
7.
16x2 = 81
13.
6z2 − 183 = 111
Challenge
2.
y2 = 121
8.
25t2 = 144
14.
3x2 = 48
*19.
3.
n2 = 625
9.
4x2 − 49 = 0
15.
4y2 = 14
2(4x + 7)
_________
= x2 + 3
4.
t2 = 12
10.
3y2 − 27 = 0
16.
5z2 = 56
*20.
15m2 − 6 = 28.5
_____
5.
x = 96
11.
5m − 140 = 40
17.
4x + 37 = 127
*21.
x2 = 36y
6.
y2 = 196
12.
9x2 + 50 = 219
18.
2(6n2 − 17) = 19
2
2
2
2
7
4
Solve.
22.
Estimate to the nearest hundredth of a second
the time it would take a car to travel 50 feet from
a stop, at a constant acceleration of 3.2 ft/s2.
1 2
Use d = __
2 at , where d represents distance, a
represents acceleration, and t represents time in
seconds.
23.
The area of a circle is 20 square inches. Estimate
the radius to the nearest tenth of an inch.
24.
The surface area of a cube is 100 square
centimeters. Estimate to the nearest tenth of a
centimeter the side length of the cube using the
formula S = 6s2, where S represents surface area
and s represents side length.
25.
26.
The height of a dropped object can by modeled
1 2
by the formula h(t) = −__
2 gt + h0 where h
represents the height after time t (in seconds), h0
represents the initial height, and g represents the
acceleration due to gravity.
A. Estimate to the nearest hundredth of a second
the time it takes an object dropped from an
initial height of 63 meters to reach the ground
on earth (g = 9.8 meters per square second).
B. Estimate to the nearest hundredth of a second
the time it takes an object dropped from an
initial height of 63 meters to reach the moon
(g = 1.62 meters per square second).
The kinetic energy of an object in motion can
1 2
be modeled by the equation E = __
2mv , where
E represents the kinetic energy, m represents
the mass, and v represents velocity. What is the
velocity (in meters per second) of an object with
a mass of 20 kilograms and kinetic energy of
33,640 Joules?
USING SQUARE ROOTS TO SOLVE EQUATIONS
359
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The Pythagorean
Theorem
Squaring and taking square roots of numbers
can help you solve many geometric problems, such
as those involving triangles.
REMEMBER
DEFINITIONS
A right angle measures 90° and is
indicated by a square □.
A triangle with a right angle is a right triangle. The two sides of the
triangle that form the right angle are the legs. The side opposite the right
angle is the hypotenuse.
hypotenuse
leg
leg
Using the Pythagorean Theorem
The Pythagorean theorem states the relationship between the lengths of the
sides of a right triangle.
PROPERTY: THE PYTHAGOREAN THEOREM
In a right triangle, the sum of the squares of the lengths of the legs equals
the square of the length of the hypotenuse.
c
a
a2 + b2 = c2
b
360
UNIT 9
RATIONALS, IRRATIONALS, AND RADICALS
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Example 1
Find the value of x.
A.
48 in.
14 in.
x in.
Solution The unknown side is the hypotenuse c.
a2 + b2 = c2
14 + 48 = x
2
2
2
196 + 2304 = x
2
THINK ABOUT IT
Substitute 14 and 48 for a and b, and x for c.
You would get the same answer
by substituting 48 for a and 14
for b.
Simplify the left side.
2500 = x2
_____
±√2500 = x
Take the square root of each side.
±50 = x
Disregard the negative answer because lengths must be nonnegative. The
value of x is 50. ■
B.
13 m
xm
12 m
Solution The unknown side is a leg.
a2 + b2 = c2
x2 + 122 = 132
Substitute x for a, 12 for b, and 13 for c.
x2 + 144 = 169
Evaluate powers.
x2 = 25
___
x = ±√25
TIP
The hypotenuse is always the
longest side of a right triangle.
You know you have made an
error if the length of a leg is
greater than the length of
the hypotenuse.
Subtract 144 from each side.
Take the square root of each side.
x = ±5
Disregard the negative answer. The value of x is 5. ■
Using the Converse of the Pythagorean Theorem
The Pythagorean theorem tells you how to find the length of a side of a
triangle given the triangle is a right triangle. The converse of the Pythagorean
theorem tells you how to determine if a triangle is a right triangle given the
lengths of all three sides.
THE CONVERSE OF THE PYTHAGOREAN THEOREM
If the sum of the squares of the lengths of the shorter sides of a triangle
equals the square of the length of the longest side, then the triangle is a
right triangle.
(continued)
THE PYTHAGOREAN THEOREM
361
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Example 2
triangle.
A.
Determine if a triangle with the given side lengths is a right
15 ft, 32 ft, and 36 ft
Solution
Substitute 15 for a, 32 for b, 36 for c, and simplify.
a + b = c2
2
2
152 + 322 362
225 + 1024 1296
1249 ≠ 1296
The two sides of the equation are not equal, so the side lengths do not form
a right triangle. ■
B.
20 ft, 16 ft, and 12 ft
Solution
Substitute 20 for c, 16 for a, 12 for b, and simplify.
a2 + b2 = c2
162 + 122 202
256 + 144 400
400 = 400 The two sides of the equation are equal. The side lengths form a right
triangle. ■
Determining the Relative Measure of the Greatest Angle
The greatest angle in a triangle is opposite the longest side c.
PROPERTY
TIP
The inequalities c > a and c > b
mean that c is the longest side in
the triangle.
For a triangle with side lengths of a, b, and c, where c > a and c > b,
• if c2 > a2 + b2, then the angle opposite side c has a measure greater
than 90°.
• if c2 < a2 + b2, then the angle opposite side c has a measure less than 90°.
Example 3 The side lengths of a triangle are 9 m, 12 m, and 14 m. Does the
greatest angle in the triangle measure 90°, more than 90°, or less than 90°?
Solution
Determine if c2 is less than, greater than, or equal to a2 + b2.
c2
a2 + b2
142
92 + 122
196
81 + 144
196 < 225
Because c2 < a2 + b2, the angle opposite side c, which is the greatest angle,
has a measure that is less than 90°. ■
362
UNIT 9
RATIONALS, IRRATIONALS, AND RADICALS
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Application: Boating and Kites
Example 4 A boat leaves a dock and travels 3.5 miles due north and
6 miles due west. How far is the boat from the dock? Round your answer to
the nearest tenth of a mile.
Solution Draw a diagram.
REMEMBER
The shortest distance between
two points is a straight line.
6 mi
TIP
3.5 mi
x mi
North
West
The unknown distance x is the length of the hypotenuse of a right triangle.
Use the Pythagorean theorem.
East
South
a2 + b2 = c2
3.52 + 62 = x2
12.25 + 36 = x2
48.25 = x2
_____
±√48.25 = x
±6.946 ≈ x
Disregard the negative answer. The boat is about 6.9 miles from the dock. ■
Example 5 Eduardo is flying a kite on a string that is 75 meters long. The
kite is 62 meters above the ground. Estimate the distance between Eduardo
and the spot on the ground directly beneath the kite.
Solution Draw a diagram.
75 m
62 m
xm
The unknown distance x is the length of a leg of a right triangle. Use the
Pythagorean theorem.
a2 + b2 = c2
x2 + 622 = 752
x2 + 3844 = 5625
x2 = 1781
_____
x = ±√1781 ≈ ±42.2
Disregard the negative answer. Eduardo is about 42.2 meters from the spot
beneath the kite. ■
THE PYTHAGOREAN THEOREM
363
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Problem Set
Find the value of x. If necessary, round your answer to the nearest
hundredth.
6.
1.
x
4
267
x
3
2.
15
12
11
7.
x
3.
x
7
x
60
4
8.
32
4.
12
x
x
55
48
8
5.
4
x
7
364
UNIT 9
RATIONALS, IRRATIONALS, AND RADICALS
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VHS_ALG_S1.indb 364
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Determine if a triangle with the given side lengths is a right triangle.
10 in., 24 in., and 26 in.
13.
11 in., 60 in., and 61 in.
10.
18 cm, 24 cm, and 30 cm
14.
51 m, 68 m, and 85 m
11.
12 ft, 20 ft, and 24 ft
15.
40 yd, 56 yd, and 70 yd
12.
20 mm, 24 mm, and 36 mm
9.
For a triangle with the given side lengths, determine if the measure of
its largest angle is greater than, less than, or equal to 90°.
16.
2 yd, 3 yd, and 4 yd
19.
5 mi, 18 mi, and 20 mi
17.
40 cm, 42 cm, and 58 cm
20.
65 km, 72 km, and 97 km
18.
21 m, 23 m, and 25 m
21.
96 ft, 140 ft, and 165 ft
26.
Mr. Edwards has placed a 20-foot flagpole in his
front yard. To help secure the pole, he has strung
wires from 2 feet below the top of the flagpole
to the ground. The wires are 19 feet long. About
how far from the bottom of the flagpole do they
reach the ground? Round to the nearest foot.
27.
Taro’s garden is in the shape of a square with an
1
area of 72 __
4 square yards. She has made a
walkway in a straight line from one corner
to the opposite corner. About how long is the
walkway? Round to the nearest yard.
28.
Jessamyn is in a hot air balloon 63 meters from
the ground. A telephone pole is directly below
her. An oak tree stands 60 meters from the
telephone pole. How far apart are the hot air
balloon and the top of the oak tree?
*29.
Challenge A boat is out to sea due east of a
marina on the shore. The marina stands exactly
halfway between two houses to the north and
south, which are 72 miles apart. The boat is
39 miles from each house. How far is the boat
from the marina?
*30.
Challenge Ashley rode her bike 6 kilometers
due south, turned and rode 4 kilometers
due east, and then turned again and rode 8
kilometers due south. To the nearest hundredth
of a kilometer, how far is she from her starting
point?
For each problem, make a sketch and solve.
22.
Southville is 57 miles due south of Portland.
A. Westfield is 76 miles due west of Portland.
How far apart are Southville and Westfield?
B. Eastborough is 27 miles due east of South-
ville. About how far is Eastborough from
Portland? Round to the nearest mile.
23.
Kenji and Isabel are standing exactly opposite
one another on either side of the bank of a
40-meter-wide river.
A. Kenji turns and walks 9 meters along the
river. If Isabel is still standing in her original
position, how far apart are Kenji and
Isabel now?
B. Isabel turns and walks 13 meters in the op-
posite direction, also along the river. About
how far apart is Isabel from Kenji’s original
position? Round to the nearest meter.
24.
Min is building a ramp from the ground to her
front doorstep. The top of the doorstep is 3.5 feet
above the ground, and the ramp is 9.5 feet long.
How far will it extend from the house? Round to
the nearest tenth of a foot.
25.
Hans leaned a 17-foot ladder against his house.
He placed the base of the ladder 8 feet from
his house, and the top of the ladder reached the
base of his window. How high is the base of his
window from the ground?
THE PYTHAGOREAN THEOREM
365
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Higher Roots
The opposite of squaring is taking a square root.
Similarly, the opposite of raising to any power n is
taking the nth root.
DEFINITION
For all a and b, if an = b, and n is an integer greater than 1, a is the nth root
of b.
NOTATION
n
__
The expression √b indicates the
principal nth root of b when n is
an even number.
PROPERTIES OF EVEN AND ODD ROOTS
n
__
Even roots have two real answers. If n is even and
b is nonnegative, √b
n __
indicates the principal, or positive, root and − √ b indicates the negative
root.
n
__
Odd roots have one real answer. If n is odd and b is positive or negative, √ b
indicates the only root, which may be positive or negative.
For instance, 3 is the third root, or cube root, of 27 because 3 · 3 · 3 = 33 = 27.
Just as 9 is a perfect square, 27 is a perfect cube.
Although you cannot take the square root of a negative number, you can
take an odd root of a negative number.____
The cube root of −27 is −3 because
3
(−3)3 = −27. This can be written as √−27 = −3.
Evaluating nth Roots
Example 1
A.
4
Evaluate each radical expression.
___
√ 81
Solution
Both 34 and (−3)4 equal 81, but the principal fourth root is 3, so
___
4
√ 81 = 3. ■
B.
6
___
√ −1
Solution
5
A negative number does not have any even roots. ■
_______
√100,000
_______
5
Solution √100,000 = 10 because 105 = 100,000. ■
_____
C.
D.
3
√ −125
Solution
366
UNIT 9
3
_____
√ −125 = −5 because (−5)3 = −125. ■
RATIONALS, IRRATIONALS, AND RADICALS
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Simplifying nth Roots
PROPERTIES
If b is a perfect nth power, then b can be factored so that the exponent on
each factor is n.
If b is not a perfect nth power, then the nth root of b is irrational.
Variable expressions can also be perfect nth powers. Assume all variables
are nonnegative.
PROPERTY
For all a__and__b when
n is odd, and for all nonnegative a and b when n is
n
n
n ___
√
√
√
even, a · b = ab when n is an integer greater than 1.
Example 2 Simplify each expression. Assume all variables are nonnegative.
3
___
√ 54
A.
Solution
3
___
3
_____
√ 54 = √ 27 · 2
3
___
3
REMEMBER
__
= √27 · √2
3
__
3
Assuming
__ a is positive when n is
n
even, √an = a.
__
= √33 · √2
3
__
= 3√2 ■
B.
___
____
√5y2 · √25y4
3
3
Solution
___
____
_____
√5y2 · √25y4 = √125y6
3
3
3
3
____
__
= √125 · √y6
3
__
3
____
= √53 · √(y2)3
3
= 5y2 ■
Simplifying Expressions with Fractional Exponents
An exponent does not have to be an integer.
PROPERTY
1
__
n
__
n
√
For all a when
___ n isnodd,
__ m and for all nonnegative a when n is even, a = a
n
m
__
m
n
and a = √ a = ( √a ) when n and m are integers greater than or equal
to 1.
(continued)
HIGHER ROOTS
367
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Example 3
Simplify each expression.
1
__
A.
42
1
__
__
2
42 = √ 4 = 2 ■
Solution
2
__
B.
(−27)3
2
__
____
2
(−27)3 = ( √−27 ) = (−3)2 = 9 ■
Solution
3
1
__
−6254
C.
1
__
___
4
____
−6254 = −( √625 ) = −(5) = −5 ■
Solution
4
√ x14
D.
Solution
___
____
4
4
x12x2
√x14 = √____
4
__
3 4
= √(x__
) · √x2
4
4
= x3√x2
2
__
= x 3x 4
1
__
= x3x 2
__
= x3 √ x ■
Rationalizing Denominators with nth Roots
When rationalizing a denominator with an nth root, be sure to multiply both
the numerator and denominator by the radical expression that will make the
radicand a perfect nth power.
Example 4 Simplify.
REMEMBER
A.
An expression with a radical
in the denominator is
not simplified.
Solution
3__
___
3
√9
3__
___
3
√9
3
__
3
√__
3__ ___
___
=3
·3
√9
__
3
√3
3
3√
___
= ____
3
√ 27
__
3
3√3
____
= 3
3
__
= √3 ■
B.
1
____
___
5
√ 16
Solution
1
____
___
5
√ 16
5
__
2
√__
1 ___
____
___
=5
√ 16
·5
__
5
√2
2
√___
= ____
5
√ 32
5
__
√2
___
= 2 ■
368
UNIT 9
RATIONALS, IRRATIONALS, AND RADICALS
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Problem Set
Simplify.
1.
2.
3.
4.
3
___
√ 64
5
9.
6.
7.
8.
____
√ −64
____
__
1
√ −32
10.
83
4__
___
11.
162
4
√8
5
____
___
12.
3
√ 16
1
__
5.
6
(−512)3
10
_____
____
4
√ 125
3
4
4
____
√ 243
2562
18.
6
____
___
19.
7__
___
20.
3
____
___
21.
18
_______
______
______
√10,000
____
√ −81
__
1
14.
(−27)3
15.
643
16.
−2163
_____
√ −125
5
13.
1
__
22.
__
2
1
__
17.
23.
5
√ 16
√3
3
√ 49
5
√10,000
3
5
25.
5
___
√x12
1
__
3
_____
√ 24x10
7
26.
27.
____
√ 486
Simplify. Assume all variables are nonnegative.
24.
___
√ 16
___
√x32
3
28.
______
5
____
5
______
√ 25x2 · √ 250x17
√ 192x11
Solve.
29.
The area of a square-shaped garden is 98 ft2.
Find the length of each side of the garden.
Express your answer in simplest radical form.
The formula for the area of a square is A = s2.
*30.
Challenge The volume of a cube-shaped hat
box is 24 in3. Find the length of each side of the
hat box. Express your answer in simplest radical
form. The formula for the volume of a cube is
V = s3.
HIGHER ROOTS
369
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