Download 8.3 Zero, Negative, and Fractional Exponents

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Chapter 8. Exponents and Polynomials
8.3 Zero, Negative, and Fractional Exponents
Learning Objectives
•
•
•
•
Simplify expressions with zero exponents.
Simplify expressions with negative exponents.
Simplify expression with fractional exponents.
Evaluate exponential expressions.
Introduction
There are many interesting concepts that arise when contemplating the product and quotient rule for exponents.
You may have already been wondering about different values for the exponents. For example, so far we have only
considered positive, whole numbers for the exponent. So called natural numbers (or counting numbers) are easy
to consider, but even with the everyday things around us we think about questions such as is it possible to have a
negative amount of money? or what would one and a half pairs of shoes look like? In this lesson, we consider what
happens when the exponent is not a natural number. We will start with What happens when the exponent is zero?
Simplify Expressions with Exponents of Zero
Let us look again at the quotient rule for exponents (that
take the example of x4 divided by x4 .
xn
xm
= xn−m ) and consider what happens when n = m. Lets
x4
= x(4−4) = x0
x4
Now we arrived at the quotient rule by considering how the factors of x cancel in such a fraction. Lets do that again
with our example of x4 divided by x4 .
x4 x · x · x · x
=
=1
x4 x · x · x · x
So x0 = 1.
This works for any value of the exponent, not just 4.
xn
= xn−n = x0
xn
Since there is the same number of factors in the numerator as in the denominator, they cancel each other out and we
obtain x0 = 1. The zero exponent rule says that any number raised to the power zero is one.
Zero Rule for Exponents: x0 = 1, x 6= 0
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8.3. Zero, Negative, and Fractional Exponents
Simplify Expressions With Negative Exponents
Again we will look at the quotient rule for exponents (that
m > n. Lets take the example of x4 divided by x6 .
x4
x6
xn
xm
= xn−m ) and this time consider what happens when
= x(4−6) = x−2 for x 6= 0.
By the quotient rule our exponent for x is −2. But what does a negative exponent really mean? Lets do the same
calculation long-hand by dividing the factors of x4 by the factors of x6 .
x4
x·x·x·x
1
1
= =
= 2
6
x
x·x x
x · x · x · x · x · x
So we see that x to the power −2 is the same as one divided by x to the power +2. Here is the negative power rule
for exponents.
Negative Power Rule for Exponents
1
xn
= x−n x 6= 0
You will also see negative powers applied to products and fractions. For example, here it is applied to a product.
(x3 y)−2 = x−6 y−2
1 1
1
x−6 y−2 = 6 · 2 = 6 2
x y
x y
using the power rule
using the negative power rule separately on each variable
Here is an example of a negative power applied to a quotient.
a −3
b
a−3
b−3
1 b3
·
a3 1
b3
a3
a−3
b−3
a−3 1
1 b3
=
· −3 = 3 ·
1 b
a 1
b3
= 3
a
3
b
=
a
=
using the power rule for quotients
using the negative power rule on each variable separately
simplifying the division of fractions
using the power rule for quotients in reverse.
The last step is not necessary but it helps define another rule that will save us time. A fraction to a negative power is
flipped.
−n
n
Negative Power Rule for Fractions xy
= xy , x 6= 0, y 6= 0
In some instances, it is more useful to write expressions without fractions and that makes use of negative powers.
Example 1
Write the following expressions without fractions.
(b)
1
x
2
x2
(c)
x2
y3
(d)
3
xy
(a)
Solution
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Chapter 8. Exponents and Polynomials
We apply the negative rule for exponents
(b)
1
−1
x =x
2
= 2x−2
x2
(c)
x2
y3
= x2 y−3
(d)
3
xy
= 3x−1 y−1
(a)
1
xn
= x−n on all the terms in the denominator of the fractions.
Sometimes, it is more useful to write expressions without negative exponents.
Example 2
Write the following expressions without negative exponents.
(a) 3x−3
(b) a2 b−3 c−1
(c) 4x−1 y3
(d)
2x−2
y−3
Solution
We apply the negative rule for exponents
(b) a2 b−3 c−1 =
=
a2
b3 c
4y3
x
(c) 4x−1 y3 =
2x−2
y−3
= x−n on all the terms that have negative exponents.
3
x3
(a) 3x−3 =
(d)
1
xn
2y3
x2
Example 3
Simplify the following expressions and write them without fractions.
(a)
4a2 b3
2a5 b
(b)
x
3y2
3
2
· x4y
Solution
(a) Reduce the numbers and apply quotient rule on each variable separately.
4a2 b3
= 2 · a2−5 · b3−1 = 2a−3 b2
6a5 b
(b) Apply the power rule for quotients first.
2x
y2
3
·
x2 y 8x2 x2 y
= 6 ·
4
y
4
Then simplify the numbers, use product rule on the x0 s and the quotient rule on the y0 s.
8x3 x2 y
·
= 2 · x3+2 · y1−6 = 2x5 y−5
y6 4
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8.3. Zero, Negative, and Fractional Exponents
Example 4
Simplify the following expressions and write the answers without negative powers.
−2 2
(a) abb3
(b)
x−3 y2
x2 y−2
Solution
(a) Apply the quotient rule inside the parenthesis.
ab−2
b3
2
= (ab−5 )2
Apply the power rule.
(ab−5 )2 = a2 b−10 =
a2
b10
(b) Apply the quotient rule on each variable separately.
x−3 y2
y4
−3−2 2−(−2)
−5 4
=
x
y
=
x
y
=
x2 y−2
x5
Simplify Expressions With Fractional Exponents
The exponent rules you learned in the last three sections apply to all powers. So far we have only looked at positive
and negative integers. The rules work exactly the same if the powers are fractions or irrational numbers. Fractional
exponents are used to express the taking of roots and radicals of something (square roots, cube roots, etc.). Here is
an exmaple.
√
1
√
√
5
2
a = a1/2 and 3 a = a1/3 and a2 = a2 5 = a 5 = a2/5
√
m
Roots as Fractional Exponents an = an/m
We will examine roots and radicals in detail in a later chapter. In this section, we will examine how exponent rules
apply to fractional exponents.
Evaluate Exponential Expressions
When evaluating expressions we must keep in mind the order of operations. You must remember PEMDAS.
Evaluate inside the Parenthesis.
Evaluate Exponents.
Perform Multiplication and Division operations from left to right.
Perform Addition and Subtraction operations from left to right.
Example 6
Evaluate the following expressions to a single number.
(a) 50
(b) 72
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(c)
(d)
Chapter 8. Exponents and Polynomials
2 3
3
3−3
Solution
(a) 50 = 1 Remember that a number raised to the power 0 is always 1.
(b) 72 = 7 · 7 = 49
3
3
8
(c) 23 = 233 = 27
(d) 3−3 =
1
33
=
1
27
Example 7
Evaluate the following expressions to a single number.
(a) 3 · 55 − 10 · 5 + 1
(b)
(c)
2·42 −3·52
32
3 −2
3
· 34
22
Solution
(a) Evaluate the exponent.
3 · 52 − 10 · 6 + 1 = 3 · 25 − 10 · 5 + 1
Perform multiplications from left to right.
3 · 25 − 10 · 5 + 1 = 75 − 50 + 1
Perform additions and subtractions from left to right.
75 − 50 + 1 = 26
(b) Treat the expressions in the numerator and denominator of the fraction like they are in parenthesis.
(2 · 42 − 3 · 52 ) (2 · 16 − 3 · 25) (32 − 75) −43
=
=
=
(32 − 22 )
(9 − 4)
5
5
(c)
3 −2
3
22
· 34 =
Example 8
2 2
2
33
· 34 =
24
36
· 34 =
24
36
· 232 =
22
35
=
4
243
Evaluate the following expressions for x = 2,y = −1, z = 3.
(a) 2x2 − 3y3 + 4z
(b) (x2 − y2 )2
2 5 −2
(c) 3x4zy
Solution
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8.3. Zero, Negative, and Fractional Exponents
(a) 2x2 − 3y+ 4z = 2 · 22 − 3 · (−1)3 + 4 · 3 = 2 · 4 − 3 · (−1) + 4 · 3 = 8 + 3 + 12 = 23
(b) (x2 − y2 )2 = (22 − (−1)2 )2 = (4 − 1)2 = 32 = 9
−2 −2
2 5 −2 2
·(−1)5
= 3·2 4·3
= 3·4·(−1)
=
(c) 3x 4z−y
12
−12 −2
12
=
−1 −2
1
=
1 2
−1
= (−1)2 = 1
Review Questions
Simplify the following expressions, be sure that there aren’t any negative exponents in the answer.
1. x−1 · y2
2. x−4
−3
3. xx−7
4.
x−3 y−5
z−7
2
1
2 −3
1
5. (x y )(x2 y 3 )
−2
6. ba
7. (3a−2 b2 c3 )3
8. x−3 · x3
Simplify the following expressions so that there aren’t any fractions in the answer.
9.
10.
11.
12.
13.
14.
15.
16.
a−3 (a5 )
a−6
5x6 y2
x8 y
(4ab6 )3
(ab)5
3
3x
y1/3
3x2 y3/2
xy1/2
(3x3 )(4x4 )
(2y)2
a−2 b−3
c−1
x1/2 y5/2
x3/2 y3/2
Evaluate the following expressions to a single number.
3−2
(6.2)0
8−4 · 86
1
(16 2 )3
x2 4x3 y4 4y2 if x = 2 and y = −1
a4 (b2 )3 + 2ab if a = −2 and b = 1
5x2 − 2y3 + 3z if x = 3, y = 2, and z = 4
2 −2
24. ab3
if a = 5 and b = 3
17.
18.
19.
20.
21.
22.
23.
Review Answers
2
1. yx
2. x14
3. x4
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4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
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20.
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23.
24.
Chapter 8. Exponents and Polynomials
z7
x3 y5
x5/2
y1/3
2
b 2
or ba2
a
27b6 c9
a6
1
a8
5x−2 y
1
64a−2 b 3
27x2 y−1
3xy
6x7 y−2
a−2 b−3 c
x−1 y
0.111
1
64
64
512
12
41
1.1664
401