Transcript Section 8.1

Chapter 8
Rational
Exponents,
Radicals, and
Complex
Numbers
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CHAPTER
8
Rational Exponents, Radicals,
and Complex Numbers
8.1
8.2
8.3
8.4
8.5
8.6
8.7
Radical Expressions and Functions
Rational Exponents
Multiplying, Dividing, and Simplifying
Radicals
Adding, Subtracting, and Multiplying
Radical Expressions
Rationalizing Numerators and
Denominators of Radical Expressions
Radical Equations and Problem Solving
Complex Numbers
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8.1
1.
2.
3.
4.
Radical Expressions and
Functions
Find the nth root of a number.
Approximate roots using a calculator.
Simplify radical expressions.
Evaluate radical functions.
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nth root: The number b is an nth root of a number a
if bn = a.
Evaluating nth roots
When evaluating a radical expression n a , the sign of a
and the index n will determine possible outcomes.
If a is nonnegative, then n a  b, where b  0 and
bn = a.
If a is negative and n is even, then there is no realnumber root.
If a is negative and n is odd, then n a  b , where b is
negative and bn = a.
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Example
Evaluate each root, if possible.
a. 169
Solution 169  13
b.  0.49
Solution  0.49  0.7
c.
100
Solution
100
is not a real number because there
is no real number whose square is
–100.
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continued
Evaluate each root, if possible.
d.  144
Solution  144  12
49
e.
144
Solution
f.
3
7
49
49


12
144
144
27
Solution
3
27  3
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continued
Evaluate each root, if possible.
g. 3 27
Solution 3 27  3
h.
4
81
Solution
4
81  3
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Some roots, like 3 are called irrational because we
cannot express their exact value using rational
numbers. In fact, writing 3 with the radical sign is
the only way we can express its exact value.
However, we can approximate 3 using rational
numbers.
Approximating to two decimal places: 2  1.41
Approximating to three decimal places: 2  1.414
Note: Remember that the symbol,
“approximately equal to.”

, means
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Example
Approximate the roots using a calculator or table in the
endpapers. Round to three decimal places.
a. 18
Solution
18  4.243
b.  32
Solution  32  5.657
c.
3
56
Solution
3
56  3.826
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Example
Find the root. Assume variables represent nonnegative
values.
a.
y
Solution
4
b. 36m
6
36 x10
c.
25 y 4
Solution
Solution
y4  y2
Because (y2)2 = y4.
3

6m
36m
6
Because (6m3)2 = 36m6.
36 x10 6 x5
 2
4
25 y
5y
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continued
Find the root. Assume variables represent nonnegative
values.
d.
3
e.
4
y
Solution
9
16
81x
Solution
3
y9
4
 y3
4
81x16  3x
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Example
Find the root. Assume variables represent any real
number.
a. y
Solution
14
b. 36 y
10
Solution
c. (n  3) Solution
2
7
y14  y
5
36 y10  6 y
(n  3) 2  n  3
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continued
Find the root. Assume variables represent any real
number.
12
6
49
y

7
y
49
y
Solution
d.
12
e.
3
27n
9
Solution
3
27n9
 3n3
c. 3 ( w  4)3 Solution 3 ( w  4)3  w  4
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Radical function: A function containing a radical
expression whose radicand has a variable.
Example
Given f(x) = 5 x  8, find f(3).
Solution
To find f(3), substitute 3 for x and simplify.
f  3  5  3  8  15  8  7
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Example
Find the domain of each of the following.
a. f  x   x  8
Solution Since the index is even, the radicand x  8  0
x 8
must be nonnegative.
Domain:  x x  8 , or [8, )
b. f  x   3x  9
Solution The radicand must be nonnegative. 3x  9  0
3x  9
Domain:  x x  3 , or (,3]
x3
Conclusion The domain of a radical function with an even
index must contain values that keep its radicand
nonnegative.
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Example
If you drop an object, the time (t) it takes in seconds
to fall d feet is given by t  16d . Find the time it
takes for an object to fall 800 feet.
Understand We are to find the time it takes for an
object to fall 800 feet.
Plan Use the formula t 
Execute
t
800
16
d
16
, replacing d with 800.
Replace d with 800.
t  50
Divide within the radical.
t  7.071
Evaluate the square root.
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continued
Answer It takes an object 7.071 seconds to fall 800
feet.
Check We can verify the calculations, which we will
leave to the viewer.
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