2.19: Simplifying Rational Exponents
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Simplifying Expressions with a1n
Rational exponents are another way of writing expressions with radicals. When we use rational exponents, we can apply the properties of exponents to simplify expressions.
The Power Property for Exponents says that (am)n=am⋅n when m and n are whole numbers. Let’s assume we are now not limited to whole numbers.
Suppose we want to find a number p such that (8p)3=8. We will use the Power Property of Exponents to find the value of p.
(8p)3=8
Multiple the exponents on the left.
83p=8
Write the exponent 1 on the right.
83p=81
Since the bases are the same, the exponents must be equal.
3p=1
Solve for p.
p=13
So (813)3=8. But we know also (3√8)3=8. Then it must be that 813=3√8.
This same logic can be used for any positive integer exponent n to show that a1n=n√a.
If n√a is a real number and n≥2, then
a1n=n√a
The denominator of the rational exponent is the index of the radical.
There will be times when working with expressions will be easier if you use rational exponents and times when it will be easier if you use radicals. In the first few examples, you'll practice converting expressions between these two notations.
Write as a radical expression:
- x12
- y13
- z14
Solution:
We want to write each expression in the form n√a.
a.
x12
The denominator of the rational exponent is 2, so the index of the radical is 2. We do not show the index when it is 2.
√x
b.
y13
The denominator of the exponent is 3, so the index is 3.
3√y
c.
z14
The denominator of the exponent is \4, so the index is 4.
4√z
Write as a radical expression:
- t12
- m13
- r14
- Answer
-
- √t
- 3√m
- 4√r
In the next example, we will write each radical using a rational exponent. It is important to use parentheses around the entire expression in the radicand since the entire expression is raised to the rational power.
Write with a rational exponent:
- √5y
- 3√4x
- 34√5z
Solution:
We want to write each radical in the form a1n
a.
√5y
No index is shown, so it is 2.
The denominator of the exponent will be 2.
Put parentheses around the entire expression 5y.
(5y)12
b.
3√4x
The index is 3, so the denominator of the exponent is 3. Include parentheses (4x).
(4x)13
c.
34√5z
The index is 4, so the denominator of the exponent is 4. Put parentheses only around the 5z since 3 is not under the radical sign.
3(5z)14
Write with a rational exponent:
- 7√3k
- 4√5j
- 83√2a
- Answer
-
- (3k)17
- (5j)14
- 8(2a)13
In the next example, you may find it easier to simplify the expressions if you rewrite them as radicals first.
Simplify:
- 2512
- 6413
- 25614
Solution:
a.
2512
Rewrite as a square root.
√25
Simplify.
5
b.
6413
Rewrite as a cube root.
3√64
Recognize 64 is a perfect cube.
3√43
Simplify.
4
c.
25614
Rewrite as a fourth root.
4√256
Recognize 256 is a perfect fourth power.
4√44
Simplify.
4
Simplify:
- 3612
- 813
- 1614
- Answer
-
- 6
- 2
- 2
Be careful of the placement of the negative signs in the next example. We will need to use the property a−n=1an in one case.
Simplify:
- (−16)14
- −1614
- (16)−14
Solution:
a.
(−16)14
Rewrite as a fourth root.
4√−16
Simplify.
There is no real number raised to the fourth power that gives us -16 so this is not a real number.
b.
−1614
The exponent only applies to the 16. Rewrite as a fourth root.
−4√16
Rewrite 16 as 24
−4√24
Simplify.
−2
c.
(16)−14
Rewrite using the property a−n=1an.
1(16)14
Rewrite as a fourth root.
14√16
Rewrite 16 as 24.
14√24
Simplify.
12
Simplify:
- (−64)−12
- −6412
- (64)−12
- Answer
-
- Not a real number
- −8
- 18
Simplifying Expressions with amn
We can look at amn in two ways. Remember the Power Property tells us to multiply the exponents and so (a1n)m and (am)1n both equal amn. If we write these expressions in radical form, we get
amn=(a1n)m=(n√a)m and amn=(am)1n=n√am
This leads us to the following definition.
For any positive integers m and n,
amn=(n√a)m and amn=n√am
Which form do we use to simplify an expression? We usually take the root first—that way we keep the numbers in the radicand smaller, before raising it to the power indicated.
Write with a rational exponent:
- √y3
- (3√2x)4
- √(3a4b)3
Solution:
We want to use amn=n√am to write each radical in the form amn
a.
Figure 2.19.1
b.
Figure 2.19.2
c.
Figure 2.19.3
Write with a rational exponent:
- 5√a2
- (3√5ab)5
- √(7xyz)3
- Answer
-
- a25
- (5ab)53
- (7xyz)32
Remember that a−n=1an. The negative sign in the exponent does not change the sign of the expression.
Simplify:
- 12523
- 16−32
- 32−25
Solution:
We will rewrite the expression as a radical first using the defintion, amn=(n√a)m. This form lets us take the root first and so we keep the numbers in the radicand smaller than if we used the other form.
a.
12523
The power of the radical is the numerator of the exponent, 2. The index of the radical is the denominator of the exponent, 3.
(3√125)2
Simplify.
(5)2
25
b. We will rewrite each expression first using a−n=1an and then change to radical form.
16−32
Rewrite using a−n=1an
11632
Change to radical form. The power of the radical is the numerator of the exponent, 3. The index is the denominator of the exponent, 2.
1(√16)3
Simplify.
143
164
c.
32−25
Rewrite using a−n=1an
13225
Change to radical form.
1(5√32)2
Rewrite the radicand as a power.
1(5√25)2
Simplify.
122
14
Simplify:
- 432
- 27−23
- 625−34
- Answer
-
- 8
- 19
- 1125
Simplify:
- −2532
- −25−32
- (−25)32
Solution:
a.
−2532
Rewrite in radical form.
−(√25)3
Simplify the radical.
−(5)3
Simplify.
−125
b.
−25−32
Rewrite using a−n=1an.
−(12532)
Rewrite in radical form.
−(1(√25)3)
Simplify the radical.
−(1(5)3)
Simplify.
−1125
c.
(−25)32
Rewrite in radical form.
(√−25)3
There is no real number whose square root is −25.
Not a real number.
Simplify:
- −1632
- −16−32
- (−16)−32
- Answer
-
- −64
- −164
- Not a real number
Using the Properties of Exponents to Simplify Expressions with Rational Exponents
The same properties of exponents that we have already used also apply to rational exponents. We will list the Properties of Exponents here to have them for reference as we simplify expressions.
Properties of Exponents
If a and b are real numbers and m and n are rational numbers, then
Product Property
am⋅an=am+n
Power Property
(am)n=am⋅n
Product to a Power
(ab)m=ambm
Quotient Property
aman=am−n,a≠0
Zero Exponent Definition
a0=1,a≠0
Quotient to a Power Property
(ab)m=ambm,b≠0
Negative Exponent Property
a−n=1an,a≠0
We will apply these properties in the next example.
Simplify:
- x12⋅x56
- (z9)23
- x13x53
Solution
a. The Product Property tells us that when we multiple the same base, we add the exponents.
x12⋅x56
The bases are the same, so we add the exponents.
x12+56
Add the fractions.
x86
Simplify the exponent.
x43
b. The Power Property tells us that when we raise a power to a power, we multiple the exponents.
(z9)23
To raise a power to a power, we multiple the exponents.
z9⋅23
Simplify.
z6
c. The Quotient Property tells us that when we divide with the same base, we subtract the exponents.
x13x53
To divide with the same base, we subtract the exponents.
1x53−13
Simplify.
1x43
Simplify:
- y34⋅y58
- (m9)29
- d15d65
- Answer
-
- y118
- m2
- 1d
Sometimes we need to use more than one property. In the next example, we will use both the Product to a Power Property and then the Power Property.
Simplify:
- (27u12)23
- (m23n12)32
Solution:
a.
(27u12)23
First we use the Product to a Power Property.
(27)23(u12)23
Rewrite 27 as a power of 3.
(33)23(u12)23
To raise a power to a power, we multiple the exponents.
(32)(u13)
Simplify.
9u13
b.
(m23n12)32
First we use the Product to a Power Property.
(m23)32(n12)32
To raise a power to a power, we multiply the exponents.
mn34
Simplify:
- (81n25)32
- (a32b12)43
- Answer
-
- 729n35
- a2b23
We will use both the Product Property and the Quotient Property in the next example.
Simplify:
- x34⋅x−14x−64
- (16x43y−56x−23y16)12
Solution:
a.
x34⋅x−14x−64
Use the Product Property in the numerator, add the exponents.
x24x−64
Use the Quotient Property, subtract the exponents.
x84
Simplify.
x2
b.
(16x43y−56x−23y16)12
Use the Quotient Property, subtract the exponents.
(16x63y66)12
Simplify.
(16x2y)12
Use the Product to a Power Property, multiply the exponents.
4xy12
Simplify:
- m23⋅m−13m−53
- (25m16n116m23n−16)12
- Answer
-
- m2
- 5nm14