Transcript Simplifying Radicals with Variables and Rational Exponents

CHAPTER
10
Rational Exponents, Radicals, and
Complex Numbers
10.1
10.2
10.3
10.4
Radical Expressions and Functions
Rational Exponents
Multiplying, Dividing, and Simplifying Radicals
Adding, Subtracting, and Multiplying Radical
Expressions
10.5 Rationalizing Numerators and Denominators of
Radical Expressions
10.6 Radical Equations and Problem Solving
10.7 Complex Numbers
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10.1
1.
2.
3.
4.
5.
6.
Radical Expressions and Functions
Find the nth root of a number.
Approximate roots using a calculator.
Simplify radical expressions.
Evaluate radical functions.
Find the domain of radical functions.
Solve applications involving 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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Slide 10- 3
Example
Evaluate each square root.
a. 169
Solution 169  13
49
b.
144
Solution
7
49
49


12
144
144
c.  196
Solution  196  14
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Slide 10- 4
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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Slide 10- 5
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
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Slide 10- 6
Example
Find the root. Assume variables represent nonnegative
values.
a.
y
b. 36m
c.
5
Solution
4
6
32x20
Solution
Solution
y4  y2
Because (y2)2 = y4.
3

6m
36m
6
5
32 x20  2 x4
Because (6m3)2 = 36m6.
Because (2x4)5 = 32x20.
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Slide 10- 7
Example
Find the root. Assume variables represent any real
number.
a. y
14
Solution
7
y14  y
2
Solution
(
n

3)
b.
(n  3) 2  n  3
c. 3 343x3
3
Solution
343x3  7x
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Slide 10- 8
Radical function: A function of the form f(x) =
where P is a polynomial.
n
P,
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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Slide 10- 9
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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Slide 10- 10
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  16d , replacing d with 800.
Execute
t
800
16
Replace d with 800.
t  50
Divide within the radical.
t  7.071
Evaluate the square root.
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Slide 10- 11
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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Slide 10- 12
For which square root is –12.37 the
approximation for?
a)  3.517
b)
3.517
c)  153
d)
153
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Slide 10- 13
For which square root is –12.37 the
approximation for?
a)  3.517
b)
3.517
c)  153
d)
153
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Slide 10- 14
Evaluate. 0.0004
a) 0.2
b) 0.02
c) 0.002
d) 0.0002
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Slide 10- 15
Evaluate. 0.0004
a) 0.2
b) 0.02
c) 0.002
d) 0.0002
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Slide 10- 16
Find the domain of f(x) = 4 x  16 .
a)  x x  4 , or [4, )
b)  x x  4 , or [4, )
c)  x x  4 , or (, 4]
d)  x x  4 , or (, 4]
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Slide 10- 17
Find the domain of f(x) = 4 x  16 .
a)  x x  4 , or [4, )
b)  x x  4 , or [4, )
c)  x x  4 , or (, 4]
d)  x x  4 , or (, 4]
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Slide 10- 18
10.2
Rational Exponents
1. Evaluate rational exponents.
2. Write radicals as expressions raised to rational
exponents.
3. Simplify expressions with rational number exponents
using the rules of exponents.
4. Use rational exponents to simplify radical expressions.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Rational exponents: An exponent that is a fraction.
Rational Exponents with a Numerator of 1
a1/n =
n
a ,where n is a natural number other than 1.
Note: If a is negative and n is odd, then the root is negative.
If a is negative and n is even, then there is no real number root.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 10- 20
Example
Rewrite using radicals, then simplify.
a. 491/2
b. 1251/3
c. 641/6
Solution
1/ 2
a. 49  49  7
b. 1251/ 3  3 125  5
c. 641/ 6  6 64  2
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Slide 10- 21
General Rule for Rational Exponents
a
m/ n
 a 
n
m
 a
n
m
, where a  0 and m and n are
natural numbers other than 1.
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Slide 10- 22
Example
Rewrite using radicals, then simplify, if possible.
a. 272/3
b. 2433/4
c. 95/2
Solution
a. 272/ 3  (271/ 3 )2  ( 3 27 ) 2  32  9
3/ 5
b. 243
c.
5/ 2
9
 (243 )  ( 243)
1/ 5 3
5
 33  27
3
 (9 )  ( 9)  3  243
1/ 2 5
5
5
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Slide 10- 23
Negative Rational Exponents
a
m / n

1
m/n
, where a  0, and m and n are natural
a
numbers with n  1.
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Slide 10- 24
Example
Rewrite using radicals, then simplify, if possible.
a. 251/2
b. 272/3
Solution
a. 251/ 2 
b. 27
2 / 3
1
1
1


1/ 2
25
25 5
1
 2/3 
27

1
3
27

2
1 1
 2 
3
9
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Slide 10- 25
Example
Write each of the following in exponential form.
a.
6
x
5
b.
4
 5x  2
3
Solution
a.
b.
6
4
5/ 6

x
x
5
 5x  2   5 x  2 
3
3/ 4
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Slide 10- 26
Rules of Exponents Summary
(Assume that no denominators are 0, that a and b are
real numbers, and that m and n are integers.)
Zero as an exponent:
a0 = 1, where a  0.
00 is indeterminate.
n
n
n
n
1
1
a
b
Negative exponents:

a
,
a  , a
b a
an
Product rule for exponents:
Quotient rule for exponents:
Raising a power to a power:
Raising a product to a power:
Raising a quotient to a power:
n
a m a n  a mn
a m  a n  a mn
m n
mn
a

a
 
n n
ab

a
b
 
a n
an
 b   bn
n
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Slide 10- 27
Example
Use the rules of exponents to simplify. Write the
answer with positive exponents. y 3/ 4  y 1/ 4
Solution
y
3/ 4
y
1/ 4
y
3/ 4  ( 1/ 4)
Use the product rule for exponents.
(Add the exponents.)
 y 2/ 4
Add the exponents.
 y1/ 2
Simplify the rational exponent.
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Slide 10- 28
Example
Use the rules of exponents to simplify. Write the
answer with positive exponents. y 5/ 6
y 1/ 6
Solution
y 5/ 6
y 1/ 6
y
5/ 6( 1/ 6)
Use the quotient for exponents.
(Subtract the exponents.)
 y 5/ 61/ 6
Rewrite the subtraction as addition.
y
Add the exponents.
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Slide 10- 29
Example
Perform the indicated operations. Write the result using
a radical.
6
a.
x  4 x3
Solution
a.
x  4 x3  x1/ 2  x 3/ 4
 x1/ 2  3/ 4
 x 2 / 4  3/ 4
 x5 / 4
 4 x5
x7
b. 3
x
6
7
7/6
x
x
b.
 1/ 3
3
x
x
 x 7 / 6 1/ 3
 x 7 / 62 / 6
 x5 / 6
 6 x5
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Slide 10- 30
Simplify.  x y
1/ 2

2 / 3 3/ 4
3/ 8 3/ 4
x
y
a)
3/8 1/ 4
x
y
b)
c) x
3/ 4
1/ 2
y
3/8 1/ 2
d) x y
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Slide 10- 31
Simplify.  x y
1/ 2

2 / 3 3/ 4
3/ 8 3/ 4
x
y
a)
3/8 1/ 4
x
y
b)
c) x
3/ 4
1/ 2
y
3/8 1/ 2
d) x y
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Slide 10- 32
Simplify. 3 25  3 5
a) 5
b) 25
c) 25
d) 5
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Slide 10- 33
Simplify. 3 25  3 5
a) 5
b) 25
c) 25
d) 5
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Slide 10- 34
Simplify.  8 
2 / 3
a) 4
1
b) 4
c) 4
1
d) 
4
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Slide 10- 35
Simplify.  8 
2 / 3
a) 4
1
b) 4
c) 4
1
d) 
4
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Slide 10- 36