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Section 6.3 Radical Expressions and Rational Exponents

Recall that in Subsection 6.1.3, we learned to evaluate the cube root of a number, say 83, we can type 8^(1/3) into a calculator. This suggests that 83=81/3. In this section, we will learn why this is true, and how to simplify expressions with rational exponents.
Many learners will find a review of exponent properties to be helpful before continuing with the current section. [cross-reference to target(s) "section-introduction-to-exponent-properties" missing or not unique] covers an introduction to exponent properties, and there is more in Section 5.6. The basic properties are summarized in List 5.6.13. These properties are still true and we can use them throughout this section whenever they might help.
Figure 6.3.1. Alternative Video Lesson

Subsection 6.3.1 Radical Expressions and Rational Exponents

Compare the following calculations:
99=3391/291/2=91/2+1/2=9=91=9
If we rewrite the above calculations with exponents, we have:
(9)2=9(91/2)2=9
Since 9 and 91/2 are both positive, and squaring either of them generates the same number, we conclude that:
9=91/2
We can verify this result by entering 9^(1/2) into a calculator, and we get 3. In general for any non-negative real number a, we have:
a=a1/2
Similarly, when a is non-negative all of the following are true:
a2=a1/2a3=a1/3a4=a1/4a5=a1/5
For example, when we see 161/4, that is equal to 164, which we know is 2 because 2222four times=16. How can we relate this to the exponential expression 161/4? In a sense, we are cutting up 16 into 4 equal parts. But not parts that you add together, rather parts that you multiply together.
Let’s summarize this information with a new exponent property.

Warning 6.3.3. Exponents on Negative Bases.

Some computers and calculators follow different conventions when there is an exponent on a negative base. To see an example of this, visit WolframAlpha
 1 
www.wolframalpha.com
and try entering cuberoot(-8), and then try (-8)^(1/3), and you will get different results. cuberoot(-8) will come out as 2, but (-8)^(1/3) will come out as a certain non-real complex number. Most likely, any calculator you are using does behave as in Fact 2, but you should confirm this.
With the Radicals and Rational Exponents Property, we can re-write radical expressions as expressions with rational exponents.

Example 6.3.4.

Write the radical expression 63 as an expression with a rational exponent. Then use a calculator to find its decimal approximation.
According to the Radicals and Rational Exponents Property, 63=61/3. A calculator tells us that 6^(1/3) works out to approximately 1.817.
For many examples that follow, we will not need a calculator. We will, however, need to recognize the roots in Figure 5.
Square Roots Cube Roots 4th-Roots 5th-Roots Roots of Powers of 2
1=1 13=1 14=1 15=1
4=2 83=2 164=2 325=2 4=2
9=3 273=3 814=3 83=2
16=4 643=4 164=2
25=5 1253=5 325=2
36=6 646=2
49=7 1287=2
64=8 2568=2
81=9 5129=2
100=10 102410=2
121=11
144=12
Figure 6.3.5. Small Roots of Appropriate Natural Numbers

Example 6.3.6.

Write the expressions in radical form using the Radicals and Rational Exponents Property and simplify the results.
  1. 41/2
  2. (9)1/2
  3. 161/4
  4. 641/3
  5. (27)1/3
  6. 31/231/2
Explanation.
  1. 41/2=4=2
  2. (9)1/2=9This value is non-real.
  3. Without parentheses around 16, the negative sign in this problem should be left out of the radical.
    161/4=164=2
  4. Here we will use the Negative Exponent Definition.
    641/3=1641/3=1643=14
  5. (27)1/3=273=3
  6. 31/231/2=33=33=9=3
The Radicals and Rational Exponents Property applies to variables in expressions just as much as it does to numbers.

Example 6.3.7.

Write the expressions as simplified as they can be using radicals.
  1. 2x1/2
  2. (5x)1/3
  3. (27x12)1/3
  4. (16x81y8)1/4
Explanation.
  1. Note that in this example the exponent is only applied to the x. Making this type of observation should be our first step for each of these exercises.
    2x1/2=2x1/2by the Negative Exponent Definition=2xby the Radicals and Rational Exponents Property
  2. In this exercise, the exponent applies to both the 5 and x.
    (5x)1/3=5x3by the Radicals and Rational Exponents Property
  3. We start out as with the previous exercise. As in the previous exercise, we have a choice as to how to simplify this expression. Here we should note that we do know what the cube root of 27 is, so we will take the path to splitting up the expression, using the Product to a Power Rule, before applying the root.
    (27x12)1/3=27x123
    Here we notice that 27 has a nice cube root, so it is good to break up the radical.
    =273x123=3x123
    Can this be simplified more? There are two ways to think about that. One way is to focus on the cube root and see that x4 cubes to make x12, and the other way is to convert the cube root back to a fraction exponent and use exponent properties.
    =3x4x4x43=3(x12)1/3=3x4=3x121/3=3x4
  4. We’ll use the exponent property for a fraction raised to a power.
    (16x81y8)1/4=(16x)1/4(81y8)1/4by the Quotient to a Power Rule=161/4x1/4811/4(y8)1/4by the Product to a Power Rule=161/4x1/4811/4y2=164x4814y2by the Radicals and Rational Exponents Property=2x43y2

Remark 6.3.8.

In general, it is easier to do algebra with rational exponents on variables than with radicals of variables. You should use Radicals and Rational Exponents Property to convert from rational exponents to radicals on variables only as a last step in simplifying.
The Radicals and Rational Exponents Property describes what can be done when there is a fractional exponent and the numerator is a 1. The numerator doesn’t have to be a 1 though and we need guidance for that situation.

Example 6.3.10. Guitar Frets.

On a guitar, there are 12 frets separating a note and the same note one octave higher. By moving from one fret to another that is five frets away, the frequency of the note changes by a factor of 25/12. Use the Full Radicals and Rational Exponents Property to write this number as a radical expression. And use a calculator to find this number as a decimal.
Explanation.
25/12=2512=3212
A calculator says 25/121.334. The fact that this is very close to 431.333 is important. It is part of the explanation for why two notes that are five frets apart on the same string would sound good to human ears when played together as a chord (known as a “fourth,” in music).

Remark 6.3.11.

By the Full Radicals and Rational Exponents Property, there are two ways to express am/n as a radical expression:
am/n=amnandam/n=(an)m
There are different times to use each formula. In general, use am/n=amn for variables and am/n=(an)m for numbers.

Example 6.3.12.

  1. Consider the expression 274/3. Use both versions of the Full Radicals and Rational Exponents Property to explain why Remark 11 says that with numbers, am/n=(an)m is preferred.
  2. Consider the expression x4/3. Use both versions of the Full Radicals and Rational Exponents Property to explain why Remark 11 says that with variables, am/n=amn is preferred.
Explanation.
  1. The expression 274/3 can be evaluated in the following two ways.
    274/3=2743by the first part of the Full Radicals and Rational Exponents Property=5314413=81or274/3=(273)4by the second part of the Full Radicals and Rational Exponents Property=34=81
    The calculation using am/n=(an)m worked with smaller numbers and can be done without a calculator. This is why we made the general recommendation in Remark 11.
  2. The expression x4/3 can be evaluated in the following two ways.
    x4/3=x43by the first part of Full Radicals and Rational Exponents Propertyorx4/3=(x3)4by the second part of the Full Radicals and Rational Exponents Property
    In this case, the simplification using am/n=amn is just shorter looking and easier to write. This is why we made the general recommendation in Remark 11.

Example 6.3.13.

Simplify the expressions using Fact 9.
  1. 82/3
  2. (64x)2/3
  3. (2764)2/3
Explanation.
  1. We will use the second part of the Full Radicals and Rational Exponents Property, since this expression only involves a number base (not variable).
    82/3=(83)2=22=4
  2. (64x)2/3=1(64x)2/3=1642/3x2/3=1(643)2x23=142x23=116x23
  3. In this problem the negative can be associated with either the numerator or the denominator, but not both. We choose the numerator.
    (2764)2/3=(27643)2by the second part of the Full Radicals and Rational Exponents Property=(273643)2=(34)2=(3)2(4)2=916

Subsection 6.3.2 More Expressions with Rational Exponents

To recap, here is a “complete” list of exponent and radical properties.
List 6.3.14. Complete List of Exponent Property
Product Property
anam=an+m
Power to a Power Property
(an)m=anm
Product to a Power Property
(ab)n=anbn
Quotient Property
anam=anm, as long as a0
Quotient to a Power Property
(ab)n=anbn, as long as b0
Zero Exponent Property
a0=1 for a0
Negative Exponent Property
an=1an
Negative Exponent Reciprocal Property
1an=an
Negative Exponent on Fraction Property
(xy)n=(yx)n
Radical and Rational Exponent Property
x1/n=xn
Radical and Rational Exponent Property
xm/n=(xn)m, usually for numbers
Radical and Rational Exponent Property
xm/n=xmn, usually for variables

Example 6.3.15.

Convert the following radical expressions into expressions with rational exponents, and simplify them if possible.
  1. 1x
  2. 1253
Explanation.
  1. 1x=1x1/2by the Radicals and Rational Exponents Property=x1/2by the Negative Exponent Definition
  2. 1253=1251/3by the Radicals and Rational Exponents Property=1(52)1/3=1521/3by the Power to a Power Rule=152/3=52/3by the Negative Exponent Definition
Learners of these simplifications often find it challenging, so we now include a many examples of varying difficulty.

Example 6.3.16.

Use exponent properties in List 14 to simplify the expressions, and write all final versions using radicals.
  1. 2w7/8
  2. 12y1/2
  3. (27b)2/3
  4. (8p6)5/3
  5. x3x4
  6. h1/3+h1/3+h1/3
  7. zz3
  8. q4
  9. 3(c1/2+d1/2)2
  10. 3(4k2/3)1/2
Explanation.
  1. 2w7/8=2w78by the Full Radicals and Rational Exponents Property
  2. 12y1/2=121y1/2by the Negative Exponent Definition=121yby the Full Radicals and Rational Exponents Property=12y
  3. (27b)2/3=(27)2/3(b)2/3by the Product to a Power Rule=(273)2b23by the Full Radicals and Rational Exponents Property=32b23=9b23
  4. (8p6)5/3=(8)5/3(p6)5/3by the Product to a Power Rule=(8)5/3p65/3by the Power to a Power Rule=(83)5p10by the Full Radicals and Rational Exponents Property=(2)5p10=32p10
  5. x3x4=x3/2x1/4by the Full Radicals and Rational Exponents Property=x3/2+1/4by the Product Rule=x6/4+1/4=x7/4=x74by the Full Radicals and Rational Exponents Property
  6. h1/3+h1/3+h1/3=3h1/3=3h3by the Radicals and Rational Exponents Property
  7. zz3=z1/2z1/3by the Radicals and Rational Exponents Property=z1/21/3by the Quotient Rule=z3/62/6=z1/6=z6by the Radicals and Rational Exponents Property
  8. q4=q1/4by the Radicals and Rational Exponents Property=(q1/4)1/2by the Radicals and Rational Exponents Property=q1/41/2by the Power to a Power Rule=q1/8=q8by the Radicals and Rational Exponents Property
  9. 3(c1/2+d1/2)2=3(c1/2+d1/2)(c1/2+d1/2)=3((c1/2)2+2c1/2d1/2+(d1/2)2)=3(c1/22+2c1/2d1/2+d1/22)=3(c+2c1/2d1/2+d)=3(c+2(cd)1/2+d)by the Product to a Power Rule=3(c+2cd+d)by the Radicals and Rational Exponents Property=3c+6cd+3d
  10. 3(4k2/3)1/2=3(4k2/3)1/2by the Negative Exponent Definition=341/2(k2/3)1/2by the Product to a Power Rule=341/2k2/31/2by the Power to a Power Rule=341/2k1/3=34k3by the Radicals and Rational Exponents Property=32k3

Example 6.3.17.

Use the radical properties in List 14 to simplify the following expressions, answering with positive rational exponents only (no radicals).
  1. 128y47
  2. 64z23z4
  3. 36xx35
  4. n23n34
  5. x37
Explanation.
  1. 128y47=(128y4)1/7by the Radicals and Rational Exponents Property=(128)1/7(y4)1/7by the Product to a Power Rule=1287(y4)1/7by the Radicals and Rational Exponents Property=2y4/7by the Power to a Power Rule
  2. 64z23z4=(64z2)1/3z1/4by the Radicals and Rational Exponents Property=(64)1/3(z2)1/3z1/4by the Product to a Power Rule=643(z2)1/3z1/4by the Radicals and Rational Exponents Property=4z2/3z1/4by the Power to a Power Rule=4z2/31/4by the Quotient Rule=4z8/123/12=4z5/12
  3. 36xx35=(36x)1/2x3/5by the Full Radicals and Rational Exponents Property=361/2x1/2x3/5by the Product to a Power Rule=36x1/2x3/5by the Full Radicals and Rational Exponents Property=6x1/23/5by the Quotient Rule=6x5/106/10=6x1/10=6x1/10by the Negative Exponent Definition
  4. n23n34=n2/3n3/4by the Full Radicals and Rational Exponents Property=n2/3+3/4by the Product Rule=n8/12+9/12=n17/12
  5. x37=x1/37by the Radicals and Rational Exponents Property=(x1/3)1/7by the Radicals and Rational Exponents Property=x1/21by the Power to a Power Rule
We will end a with a short application of rational exponents. Kepler’s Laws of Orbital Motion
 2 
en.wikipedia.org/wiki/Kepler%27s_laws_of_planetary_motion
describe how planets orbit stars and how satellites orbit planets. In particular, his third law has a rational exponent, which we will now explore.

Example 6.3.18. Kepler and the Satellite.

Kepler’s third law of motion says that for objects with a roughly circular orbit that the time (in hours) that it takes to make one full revolution around the planet, T, is proportional to three-halves power of the distance (in kilometers) from the center of the planet to the satellite, r. For the Earth, it looks like this:
T=2πGMEr3/2
In this case, both G and ME are constants. G stands for the universal gravitational constant
 3 
en.wikipedia.org/wiki/Gravitational_constant
where G is about 8.65×1013 km3kg·h2 and ME stands for the mass of the Earth
 4 
en.wikipedia.org/wiki/Earth_mass
where ME is about 5.972×1024 kg. Inputting these values into this formula yields a simplified version that looks like this:
T2.76×106r3/2
Most satellites orbit in what is called low Earth orbit
 5 
en.wikipedia.org/wiki/Low_Earth_orbit
, including the international space station which orbits at about 340 km above from Earth’s surface. The Earth’s average radius is about 6380 km. Find the period of the international space station.
Explanation.
The formula has already been identified, but the input takes just a little thought. The formula uses r as the distance from the center of the Earth to the satellite, so to find r we need to combine the radius of the Earth and the distance to the satellite above the surface of the Earth.
r=340+6380=6720
Now we can input this value into the formula and evaluate.
T2.76106r3/22.76106(6720)3/22.76106(6720)31.52
The formula tells us that it takes a little more than an hour and a half for the ISS to orbit the Earth! That works out to 15 or 16 sunrises per day.

Reading Questions 6.3.3 Reading Questions

1.

Raising a number to a reciprocal power (like 12 or 15) is the same as doing what other thing to that number?

2.

When the exponent on an expression is a fraction like 35, which part of the fraction is essentially the index of a radical?

Exercises 6.3.4 Exercises

Review and Warmup

Exercise Group.
Use the properties of exponents to simplify the expression. If there are any exponents in your answer, they should be positive.
1.
x19x20
2.
r2r14
3.
(x3)5
4.
(y4)12
5.
(9x510)3
6.
(5x64)2
7.
(4t8)3
8.
(10r9)3
9.
r17r4
10.
t19t16
11.
x12x5
12.
x6x5
13.
(4y17)(9y4)
14.
(2y11)(5y7)

Skills Practice

Exercise Group.
Evaluate each expression without help from a calculator.
15.
8112
16.
2723
17.
(181)34
18.
(19)32
19.
3245
20.
2532
21.
10245
22.
643
23.
8273
24.
271253
25.
27643
26.
271253
27.
(a)
8112
(b)
(81)12
(c)
8112
28.
(a)
10012
(b)
(100)12
(c)
10012
29.
(a)
813
(b)
(8)13
(c)
813
30.
(a)
2713
(b)
(27)13
(c)
2713
31.
(a)
83
(b)
83
(c)
83
32.
(a)
273
(b)
273
(c)
273
33.
(a)
164
(b)
164
(c)
164
34.
(a)
814
(b)
814
(c)
814
Exercise Group.
Use a calculator to approximate the expression with a decimal to four significant digits.
35.
625
36.
1747
37.
1934
38.
923
Convert Radicals to Fractional Exponents.
Write the expression with rational exponents.
39.
y8
40.
z5
41.
2t+94
42.
8r+33
43.
m5
44.
n
45.
1b47
46.
1c54
Convert Fractional Exponents to Radicals.
Convert the expression to use radical notation.
47.
x23
48.
y45
49.
z43
50.
r23
51.
215r45
52.
1014m34
53.
n58
54.
b58
55.
c27
56.
x56
57.
217y47
58.
218z58
Simplifying Expressions with Rational Exponents.
Simplify the expression, answering with rational exponents and not radicals.
59.
t11t11
60.
r7r7
61.
4m
62.
32n45
63.
25aa310
64.
100cc56
65.
27x43x56
66.
27y43y56
67.
z5z310
68.
tt56
69.
r34
70.
m5
71.
nn8
72.
tt9

Applications

Exercise Group.
On a guitar, there are 12 frets separating a note and the same note one octave higher.
73.
By moving from one fret to another that is seven frets away, the frequency of the note changes by a factor of 27/12. Use a calculator to find this number as a decimal.
This decimal shows you that 27/12 is very close to a “nice” fraction with a small denominator. Two notes with this frequency ratio form a “perfect fifth” in music. What is that fraction?
74.
By moving from one fret to another that is four frets away, the frequency of the note changes by a factor of 24/12. Use a calculator to find this number as a decimal.
This decimal shows you that 24/12 is very close to a “nice” fraction with a small denominator. Two notes with this frequency ratio form a “major third” in music. What is that fraction?
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