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Properties of Exponents
Eve Rawley, (EveR)
Anne Gloag, (AnneG)
Andrew Gloag, (AndrewG)
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Printed: November 19, 2014
AUTHORS
Eve Rawley, (EveR)
Anne Gloag, (AnneG)
Andrew Gloag, (AndrewG)
www.ck12.org
Chapter 1. Properties of Exponents
C HAPTER
1
Properties of Exponents
C HAPTER O UTLINE
1.1
Exponent Properties Involving Products
1.2
Exponent Properties Involving Quotients
1.3
Zero, Negative, and Fractional Exponents
1
1.1. Exponent Properties Involving Products
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1.1 Exponent Properties Involving Products
Learning Objectives
• Use the product of a power property.
• Use the power of a product property.
• Simplify expressions involving product properties of exponents.
Introduction
Back in chapter 1, we briefly covered expressions involving exponents, like 35 or x3 . In these expressions, the
number on the bottom is called the base and the number on top is the power or exponent. The whole expression is
equal to the base multiplied by itself a number of times equal to the exponent; in other words, the exponent tells us
how many copies of the base number to multiply together.
Example 1
Write in exponential form.
a) 2 · 2
b) ( 3)( 3)( 3)
c) y · y · y · y · y
d) (3a)(3a)(3a)(3a)
Solution
a) 2 · 2 = 22 because we have 2 factors of 2
b) ( 3)( 3)( 3) = ( 3)3 because we have 3 factors of (-3)
c) y · y · y · y · y = y5 because we have 5 factors of y
d) (3a)(3a)(3a)(3a) = (3a)4 because we have 4 factors of 3a
When the base is a variable, it’s convenient to leave the expression in exponential form; if we didn’t write x7 , we’d
have to write x · x · x · x · x · x · x instead. But when the base is a number, we can simplify the expression further than
that; for example, 27 equals 2 · 2 · 2 · 2 · 2 · 2 · 2, but we can multiply all those 2’s to get 128.
Let’s simplify the expressions from Example 1.
Example 2
Simplify.
a) 22
b) ( 3)3
c) y5
d) (3a)4
Solution
a) 22 = 2 · 2 = 4
2
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Chapter 1. Properties of Exponents
b) ( 3)3 = ( 3)( 3)( 3) = 27
c) y5 is already simplified
d) (3a)4 = (3a)(3a)(3a)(3a) = 3 · 3 · 3 · 3 · a · a · a · a = 81a4
Be careful when taking powers of negative numbers. Remember these rules:
(negative number) · (positive number) = negative number
(negative number) · (negative number) = positive number
So even powers of negative numbers are always positive. Since there are an even number of factors, we pair up the
negative numbers and all the negatives cancel out.
( 2)6 = ( 2)( 2)( 2)( 2)( 2)( 2) = ( 2)( 2) · ( 2)( 2) · ( 2)( 2) = +64
| {z } | {z } | {z }
+4
+4
+4
And odd powers of negative numbers are always negative. Since there are an odd number of factors, we can still
pair up negative numbers to get positive numbers, but there will always be one negative factor left over, so the answer
is negative:
( 2)5 = ( 2)( 2)( 2)( 2)( 2) = ( 2)( 2) · ( 2)( 2) · ( 2) = 32
| {z } | {z } | {z }
+4
+4
2
Use the Product of Powers Property
So what happens when we multiply one power of x by another? Let’s see what happens when we multiply x to the
power of 5 by x cubed. To illustrate better, we’ll use the full factored form for each:
(x · x · x · x · x) · (x · x · x) = (x · x · x · x · x · x · x · x)
|
{z
} | {z } |
{z
}
x5
x3
x8
So x5 ⇥ x3 = x8 . You may already see the pattern to multiplying powers, but let’s confirm it with another example.
We’ll multiply x squared by x to the power of 4:
(x · x) · (x · x · x · x) = (x · x · x · x · x · x)
| {z } | {z } |
{z
}
x2
x4
x6
So x2 ⇥ x4 = x6 . Look carefully at the powers and how many factors there are in each calculation. 5 x’s times 3 x’s
equals (5 + 3) = 8 x’s. 2 x’s times 4 x’s equals (2 + 4) = 6 x’s.
You should see that when we take the product of two powers of x, the number of x’s in the answer is the total number
of x’s in all the terms you are multiplying. In other words, the exponent in the answer is the sum of the exponents in
the product.
Product Rule for Exponents: xn · xm = x(n+m)
There are some easy mistakes you can make with this rule, however. Let’s see how to avoid them.
Example 3
3
1.1. Exponent Properties Involving Products
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Multiply 22 · 23 .
Solution
22 · 23 = 25 = 32
Note that when you use the product rule you don’t multiply the bases. In other words, you must avoid the common
error of writing 22 · 23 = 45 . You can see this is true if you multiply out each expression: 4 times 8 is definitely 32,
not 1024.
Example 4
Multiply 22 · 33 .
Solution
22 · 33 = 4 · 27 = 108
In this case, we can’t actually use the product rule at all, because it only applies to terms that have the same base. In
a case like this, where the bases are different, we just have to multiply out the numbers by hand—the answer is not
25 or 35 or 65 or anything simple like that.
Use the Power of a Product Property
What happens when we raise a whole expression to a power? Let’s take x to the power of 4 and cube it. Again we’ll
use the full factored form for each expression:
(x4 )3 = x4 ⇥ x4 ⇥ x4
3 f actors o f {x to the power 4}
(x · x · x · x) · (x · x · x · x) · (x · x · x · x) = x · x · x · x · x · x · x · x · x · x · x · x = x12
So (x4 )3 = x12 . You can see that when we raise a power of x to a new power, the powers multiply.
Power Rule for Exponents: (xn )m = x(n·m)
If we have a product of more than one term inside the parentheses, then we have to distribute the exponent over all
the factors, like distributing multiplication over addition. For example:
(x2 y)4 = (x2 )4 · (y)4 = x8 y4 .
Or, writing it out the long way:
(x2 y)4 = (x2 y)(x2 y)(x2 y)(x2 y) = (x · x · y)(x · x · y)(x · x · y)(x · x · y)
= x · x · x · x · x · x · x · x · y · y · y · y = x 8 y4
Note that this does NOT work if you have a sum or difference inside the parentheses! For example, (x+y)2 6= x2 +y2 .
This is an easy mistake to make, but you can avoid it if you remember what an exponent means: if you multiply out
(x + y)2 it becomes (x + y)(x + y), and that’s not the same as x2 + y2 . We’ll learn how we can simplify this expression
in a later chapter.
The following video from YourTeacher.com may make it clearer how the power rule works for a variety of exponential expressions:
http://www.youtube.com/watch?v=Mm4y_I8-hoU
4
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Chapter 1. Properties of Exponents
Example 5
Simplify the following expressions.
a) 35 · 37
b) 26 · 2
c) (42 )3
Solution
When we’re just working with numbers instead of variables, we can use the product rule and the power rule, or we
can just do the multiplication and then simplify.
a) We can use the product rule first and then evaluate the result: 35 · 37 = 312 = 531441.
OR we can evaluate each part separately and then multiply them: 35 · 37 = 243 · 2187 = 531441.
b) We can use the product rule first and then evaluate the result: 26 · 2 = 27 = 128.
OR we can evaluate each part separately and then multiply them: 26 · 2 = 64 · 2 = 128.
c) We can use the power rule first and then evaluate the result: (42 )3 = 46 = 4096.
OR we can evaluate the expression inside the parentheses first, and then apply the exponent outside the parentheses:
(42 )3 = (16)3 = 4096.
Example 6
Simplify the following expressions.
a) x2 · x7
b) (y3 )5
Solution
When we’re just working with variables, all we can do is simplify as much as possible using the product and power
rules.
a) x2 · x7 = x2+7 = x9
b) (y3 )5 = y3⇥5 = y15
Example 7
Simplify the following expressions.
a) (3x2 y3 ) · (4xy2 )
b) (4xyz) · (x2 y3 ) · (2yz4 )
c) (2a3 b3 )2
Solution
When we have a mix of numbers and variables, we apply the rules to each number and variable separately.
a) First we group like terms together: (3x2 y3 ) · (4xy2 ) = (3 · 4) · (x2 · x) · (y3 · y2 )
Then we multiply the numbers or apply the product rule on each grouping: = 12x3 y5
b) Group like terms together: (4xyz) · (x2 y3 ) · (2yz4 ) = (4 · 2) · (x · x2 ) · (y · y3 · y) · (z · z4 )
Multiply the numbers or apply the product rule on each grouping: = 8x3 y5 z5
c) Apply the power rule for each separate term in the parentheses: (2a3 b3 )2 = 22 · (a3 )2 · (b3 )2
Multiply the numbers or apply the power rule for each term = 4a6 b6
5
1.1. Exponent Properties Involving Products
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Example 8
Simplify the following expressions.
a) (x2 )2 · x3
b) (2x2 y) · (3xy2 )3
c) (4a2 b3 )2 · (2ab4 )3
Solution
In problems where we need to apply the product and power rules together, we must keep in mind the order of
operations. Exponent operations take precedence over multiplication.
a) We apply the power rule first: (x2 )2 · x3 = x4 · x3
Then apply the product rule to combine the two terms: x4 · x3 = x7
b) Apply the power rule first: (2x2 y) · (3xy2 )3 = (2x2 y) · (27x3 y6 )
Then apply the product rule to combine the two terms: (2x2 y) · (27x3 y6 ) = 54x5 y7
c) Apply the power rule on each of the terms separately: (4a2 b3 )2 · (2ab4 )3 = (16a4 b6 ) · (8a3 b12 )
Then apply the product rule to combine the two terms: (16a4 b6 ) · (8a3 b12 ) = 128a7 b18
Homework Problems
Write in exponential notation:
1.
2.
3.
4.
5.
4·4·4·4·4
3x · 3x · 3x
( 2a)( 2a)( 2a)( 2a)
6·6·6·x·x·y·y·y·y
2·x·y·2·2·y·x
Find each number.
6.
7.
8.
9.
10.
11.
54
( 2)6
(0.1)5
( 0.6)3
(1.2)2 + 53
32 · (0.2)3
Multiply and simplify:
12.
13.
14.
15.
16.
17.
63 · 66
22 · 24 · 26
32 · 43
x2 · x4
( 2y4 )( 3y)
(4a2 )( 3a)( 5a4 )
Simplify:
18. (a3 )4
6
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19.
20.
21.
22.
23.
24.
25.
Chapter 1. Properties of Exponents
(xy)2
(3a2 b3 )4
( 2xy4 z2 )5
( 8x)3 (5x)2
(4a2 )( 2a3 )4
(12xy)(12xy)2
(2xy2 )( x2 y)2 (3x2 y2 )
7
1.2. Exponent Properties Involving Quotients
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1.2 Exponent Properties Involving Quotients
Learning Objectives
• Use the quotient of powers property.
• Use the power of a quotient property.
• Simplify expressions involving quotient properties of exponents.
Use the Quotient of Powers Property
The rules for simplifying quotients of exponents are a lot like the rules for simplifying products. Let’s look at what
happens when we divide x7 by x4 :
x7 x · x · x · x · x · x · x x · x · x
=
=
= x3
4
x
x·x·x·x
1
You can see that when we divide two powers of x, the number of x’s in the solution is the number of x’s in the top of
the fraction minus the number of x’s in the bottom. In other words, when dividing expressions with the same base,
we keep the same base and simply subtract the exponent in the denominator from the exponent in the numerator.
Quotient Rule for Exponents:
xn
xm
= x(n
m)
When we have expressions with more than one base, we apply the quotient rule separately for each base:
x 5 y3 x · x · x · x · x y · y · y x · x y
=
·
=
· = x2 y
3
2
x y
x·x·x
y·y
1 1
OR
x 5 y3
= x5
3
2
x y
3
· y3
2
= x2 y
Example 1
Simplify each of the following expressions using the quotient rule.
a)
x10
x5
b)
a6
a
c)
a5 b4
a3 b2
Solution
a)
x10
x5
= x10
b)
a6
a
= a6
c)
a5 b4
a3 b2
5
1
= a5
= x5
= a5
3 · b4 2
= a2 b2
Now let’s see what happens if the exponent in the denominator is bigger than the exponent in the numerator. For
4
example, what happens when we apply the quotient rule to xx7 ?
The quotient rule tells us to subtract the exponents. 4 minus 7 is -3, so our answer is x 3 . A negative exponent!
What does that mean?
8
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Chapter 1. Properties of Exponents
Well, let’s look at what we get when we do the division longhand by writing each term in factored form:
x4
x·x·x·x
1
1
=
=
=
x7 x · x · x · x · x · x · x x · x · x x3
Even when the exponent in the denominator is bigger than the exponent in the numerator, we can still subtract the
powers. The x’s that are left over after the others have been canceled out just end up in the denominator instead of
7
3
4
the numerator. Just as xx4 would be equal to x1 (or simply x3 ), xx7 is equal to x13 . And you can also see that x13 is equal
to x 3 . We’ll learn more about negative exponents shortly.
Example 2
Simplify the following expressions, leaving all exponents positive.
a)
x2
x6
b)
a2 b6
a5 b
Solution
a) Subtract the exponent in the numerator from the exponent in the denominator and leave the x’s in the denominator:
x2
= x61 2 = x14
x6
b) Apply the rule to each variable separately:
a2 b6
a5 b
=
1
a5 2
6 1
· b1 =
b5
a3
The Power of a Quotient Property
When we raise a whole quotient to a power, another special rule applies. Here is an example:
✓
x3
y2
◆4
=
✓
x3
y2
◆ ✓ 3◆ ✓ 3◆ ✓ 3◆
x
x
x
(x · x · x) · (x · x · x) · (x · x · x) · (x · x · x) x12
· 2 · 2 · 2 =
= 8
y
y
y
(y · y) · (y · y) · (y · y) · (y · y)
y
Notice that the exponent outside the parentheses is multiplied by the exponent in the numerator and the exponent in
the denominator, separately. This is called the power of a quotient rule:
⇣ n ⌘p
n·p
Power Rule for Quotients: yxm = yxm·p
Let’s apply these new rules to a few examples.
Example 3
Simplify the following expressions.
a)
45
42
b)
53
57
c)
⇣ 4 ⌘2
3
52
Solution
Since there are just numbers and no variables, we can evaluate the expressions and get rid of the exponents completely.
45
42
= 45
1024
16
= 64
a) We can use the quotient rule first and then evaluate the result:
OR we can evaluate each part separately and then divide:
45
42
=
2
= 43 = 64
9
1.2. Exponent Properties Involving Quotients
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b) Use the quotient rule first and hen evaluate the result:
OR evaluate each part separately and then reduce:
53
57
=
53
57
1
54
=
=
1
625
=
125
78125
1
625
Notice that it makes more sense to apply the quotient rule first for examples (a) and (b). Applying the exponent rules
to simplify the expression before plugging in actual numbers means that we end up with smaller, easier numbers to
work with.
⇣ 4 ⌘2
8
c) Use the power rule for quotients first and then evaluate the result: 352 = 354 = 6561
625
⇣ 4 ⌘2
2
OR evaluate inside the parentheses first and then apply the exponent: 352 = 81
= 6561
25
625
Example 4
Simplify the following expressions:
a)
b)
x12
x5
⇣ 4 ⌘5
x
x
Solution
a) Use the quotient rule:
x12
x5
= x12
5
= x7
⇣ 4 ⌘5
x
x
x20
x5
= x15
⇣ 4 ⌘5
OR use the quotient rule inside the parentheses first, then apply the power rule: xx
= (x3 )5 = x15
b) Use the power rule for quotients and then the quotient rule:
=
Example 5
Simplify the following expressions.
a)
6x2 y3
2xy2
b)
⇣
2a3 b3
8a7 b
Solution
⌘2
When we have a mix of numbers and variables, we apply the rules to each number or each variable separately.
a) Group like terms together:
6x2 y3
2xy2
2
3
= 62 · xx · yy2
Then reduce the numbers and apply the quotient rule on each fraction to get 3xy.
⇣ 3 3 ⌘2 ⇣ 2 ⌘2
b
b
b) Apply the quotient rule inside the parentheses first: 2a
= 4a
4
8a7 b
⇣ 2 ⌘2
b
b4
Then apply the power rule for quotients: 4a
= 16a
4
8
Example 6
Simplify the following expressions.
6
a) (x2 )2 · xx4
⇣ 2 ⌘3 2
b) 16a
· ab16
4b5
Solution
In problems where we need to apply several rules together, we must keep the order of operations in mind.
a) We apply the power rule first on the first term:
10
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Chapter 1. Properties of Exponents
(x2 )2 ·
6
x6
4 x
=
x
·
x4
x4
Then apply the quotient rule to simplify the fraction:
x4 ·
x6
= x4 · x2
4
x
And finally simplify with the product rule:
x4 · x2 = x6
b)
⇣
16a2
4b5
⌘3
2
· ab16
Simplify inside the parentheses by reducing the numbers:
✓
4a2
b5
◆3
b2
· 16
a
Then apply the power rule to the first fraction:
✓
4a2
b5
◆3
·
b2
64a6 b2
=
·
a16
b15 a16
Group like terms together:
64a6 b2
a6 b2
·
=
64
·
·
b15 a16
a16 b15
And apply the quotient rule to each fraction:
64 ·
a6 b2
64
·
=
a16 b15 a10 b13
Homework Problems
Evaluate the following expressions.
1.
2.
3.
4.
56
52
67
63
34
310
3 2
8
11
1.2. Exponent Properties Involving Quotients
5.
6.
7.
8.
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⇣ 2 ⌘3
2
33
22 ·32
52
33 ·52
7
⇣3 3 2 ⌘2
2 ·4
24
Simplify the following expressions.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
12
a3
a2
x5
9
⇣x 3 4 ⌘3
a b
a2 b
6
x y2
x2 y5
6a3
2a2
15x5
⇣5x 4 ⌘4
18a
15a10
25yx6
5 2
⇣20y6 x2 ⌘3
x y
x 4 y4
⇣ 2 ⌘2
6a
· 5b
3a
4b4
(3ab)2 (4a3 b4 )3
(6a2 b)4
(2a2 bc2 )(6abc3 )
4ab2 c
(2a2 bc2 )(6abc3 )
for a = 2, b = 1, and c = 3
2
⇣ 2 4ab
⌘3c 2
3x y
· zx for x = 1, y = 2, and z = 1
2z
⇣ ⌘2
2x3
x
· 2y
for x = 2, y = 3
xy2
⇣ ⌘2
2x3
x
·
for x = 0, y = 6
2y
xy2
2
3
If a = 2 and b = 3, simplify (a ab)(bc)
as much
3 c2
as possible.
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Chapter 1. Properties of Exponents
1.3 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
x4
= x(4
4
x
= xn
4)
m)
and consider what happens when n = m. Let’s
= x0
Now we arrived at the quotient rule by considering how the factors of x cancel in such a fraction. Let’s 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
x
n
= x0
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
13
1.3. Zero, Negative, and Fractional Exponents
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Simplify Expressions With Negative Exponents
Again we will look at the quotient rule for exponents (that
m > n. Let’s take the example of x4 divided by x6 .
x4
x6
= x(4
6)
2
=x
xn
xm
= xn
m)
and this time consider what happens when
for x 6= 0.
By the quotient rule our exponent for x is -2. But what does a negative exponent really mean? Let’s do the same
calculation long-hand by dividing the factors of x4 by the factors of x6 .
x·x·x·x
1
1
x4
=
=
=
x6 x · x · x · x · x · x x · x x2
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 6y 2
1
1 1
x 6y 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⌘
b
a
b
3
3
3
1 b3
·
a3 1
b3
a3
a 3
= 3
b
a 3 1
1 b3
=
· 3= 3·
1 b
a 1
3
b
= 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
14
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Chapter 1. Properties of Exponents
We apply the negative rule for exponents
(b)
1
1
x =x
2
= 2x 2
x2
(c)
x2
y3
= x2 y
(d)
3
xy
= 3x 1 y
(a)
1
xn
=x
n
on all the terms in the denominator of the fractions.
3
1
Sometimes, it is more useful to write expressions without negative exponents.
Example 2
Write the following expressions without negative exponents.
3
(a) 3x
(b) a2 b 3 c
1
(c) 4x 1 y3
(d)
2x
y
2
3
Solution
We apply the negative rule for exponents
=
3
x3
(b) a2 b 3 c
1
=
(c) 4x 1 y3 =
4y3
x
3
(a) 3x
(d)
2x
y
2
3
=
1
xn
=x
n
on all the terms that have negative exponents.
a2
b3 c
2y3
x2
Example 3
Simplify the following expressions and write them without fractions.
(a)
(b)
4a2 b3
2a5 b
⇣
x
3y2
⌘3
2
· x4y
Solution
(a) Reduce the numbers and apply quotient rule on each variable separately.
4a2 b3
= 2 · a2
6a5 b
5
· b3
1
= 2a 3 b2
(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 x’s and the quotient rule on the y’s.
8x3 x2 y
·
= 2 · x3+2 · y1
6
y
4
6
= 2x5 y
5
15
1.3. Zero, Negative, and Fractional Exponents
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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
=x
x2 y 2
3 2 2 ( 2)
y
y4
=x y = 5
x
5 4
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.
p
1
p
p
5
1
2
2
1
a = a 2 and 3 a = a 3 and a2 = a2 5 = a 5 = a 5
p
n
m
Roots as Fractional Exponents an = a 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.
Example 5
Simplify the following expressions.
1
1
(a) a 2 · a 3
⇣ 1 ⌘2
(b) a 3
5
(c)
a2
(d)
⇣ 2 ⌘ 31
1
a2
x
y3
Solution
(a) Apply the product rule.
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Chapter 1. Properties of Exponents
1
1
1
1
5
a2 ·a3 = a2+3 = a6
(b) Apply the power rule.
⇣
a
1
3
⌘2
2
= a3
(c) Apply the quotient rule.
5
a2
a
1
2
5
= a2
1
2
4
= a 2 = a2
(d) Apply the power rule for quotients.
✓
x2
y3
◆ 13
2
x3
=
y
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
(c)
(d)
2 3
3
3 3
1
(e) 16 2
(f) 8
1
3
Solution
(a) 50 = 1 Remember that a number raised to the power 0 is always 1.
(b) 72 = 7 · 7 = 49
(c)
(d) 3
2 3
3
3
1
23
8
= 27
33
1
1
= 27
33
=
=
(e) 16 2 =
p
16 = 4 Remember that an exponent of
1
2
means taking the square root.
17
1.3. Zero, Negative, and Fractional Exponents
(f) 8
1
3
=
1
1
83
1
= p
=
3
8
1
2
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Remember that an exponent of
1
3
means taking the cube root.
Example 7
Evaluate the following expressions to a single number.
(a) 3 · 55
(b)
2·42
(c)
⇣ 3⌘
10 · 5 + 1
3·52
32
3
22
2
· 34
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
(32
(c)
⇣ 3⌘
3
22
2
3
4
· =
Example 8
⇣ 2 ⌘2
2
33
· 34 =
24
36
· 34 =
3 · 52 ) (2 · 16
=
22 )
(9
24
36
· 232 =
22
35
=
3 · 25) (32 75)
43
=
=
4)
5
5
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
(a) 2x2
3y3 + 4z = 2 · 22
3 · ( 1)3 + 4 · 3 = 2 · 4
(b) (x2 y2 )2 = (22 ( 1)2 )2 = (4 1)2 = 32 = 9
⇣ 2 5⌘ 2 ⇣ 2
⌘ 2 ⇣
⌘ 2
3·2 ·( 1)5
3·4·( 1)
3x y
(c)
=
=
=
4z
4·3
12
3 · ( 1) + 4 · 3 = 8 + 3 + 12 = 23
12
12
2
=
1
1
2
=
1 2
1
= ( 1)2 = 1
Homework Problems
Simplify the following expressions, be sure that there aren’t any negative exponents in the answer.
18
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1 · y2
1. x
2. x
3. xx
4.
x
Chapter 1. Properties of Exponents
4
3
7
3y 5
z
a
b
7
2
5.
6. (3a 2 b2 c3 )3
7. x 3 · x3
Simplify the following expressions so that there aren’t any fractions in the answer.
9.
10.
11.
12.
13.
3 (a5 )
a
a 6
5x6 y2
x8 y
(4ab6 )3
(ab)5
(3x3 )(4x4 )
(2y)2
a 2b 3
c 1
Evaluate the following expressions to a single number.
3 2
(6.2)0
8 4 · 86
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
23. ba3
if a = 5 and b = 3
17.
18.
19.
20.
21.
22.
Review Answers
2
1. yx
2. x14
3. x4
7
4. x3zy5
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
5
x2
1
y3
2
b 2
or ba2
a
27b6 c9
a6
1
a8
5x 2 y
27x2 y 1
3xy
6x7 y 2
a 2b 3c
x 1y
0.111
1
64
19
1.3. Zero, Negative, and Fractional Exponents
19.
20.
21.
22.
23.
20
64
512
12
41
1.1664
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