350
Chapter 13
◆
Exponents and Radicals
67. The power to a resistor is given by Eq. A67. If the current is then doubled and the resistance is halved, we have
R
(2 I )2 p q
2
Simplify this expression.
13–2
Unless otherwise stated, we’ll
assume that when the index is
even, the radicand is positive.
We’ll consider square roots and
fourth roots of negative numbers
when we discuss imaginary
numbers in Chapter 21.
Simplification of Radicals
Relation between Fractional Exponents and Radicals
A quantity raised to a fractional exponent can also be written as a radical. A radical consists of
a radical sign, a quantity under the radical sign called the radicand, and the index of the radical.
A radical is written as follows:
radical sign
n
兹
a
index
radicand
where n is an integer. An index of 2, for a square root, is usually not written.
In general, the following equation applies:
Relation
between
Exponents
and Radicals
◆◆◆
n
a1/n 兹
a
36
Example 12:
(a) 41/2 兹 4 2
(b) 81/3 兹8 2
(c) x1/2 兹x
1
(e) w1/2 兹w
4
(d) y1/4 兹
y
3
◆◆◆
◆◆◆
5
Example 13: Simplify 兹382 .
Solution:
兹5 382 3821/5 3.28
◆◆◆
by calculator, rounded to three significant digits.
If a quantity is raised to a fractional exponent m/n (where m and n are integers), we have
am/n (am)1/n 兹am
n
Further,
n
am/n (a1/n)m (兹
a)m
So we have the following formula:
n
am/n 兹am (兹
a)m
n
37
◆
Section 13–2
◆◆◆
351
Simplification of Radicals
Example 14:
(a) 82/3 (兹8 )2 = (2)2 4
5 (b) 兹
(68.3)3 (68.3)3/5 12.6, by calculator
3
(c) 兹x4 x4/3
5 (d) (兹
y2 )3 (y2/5)3 y6/5
3
◆◆◆
We use these definitions to switch between exponential form and radical form.
◆◆◆
Example 15: Express x1/2y1/3 in radical form.
Solution:
x1/2 兹x
x1/2y1/3 3
y1/3 兹
y
◆◆◆
◆◆◆
3 2
Example 16: Express 兹
y in exponential form.
Solution:
兹y2 (y2)1/3 y2/3
3
◆◆◆
◆◆◆
4
Example 17: Find the value of 兹
(81)3 without using a calculator.
Solution: Instead of cubing 81 and then taking the fourth root, let us take the fourth root first.
4 3
4
兹(81)3 兹81 (3)3 27
Common
Error
p
q
◆◆◆
Don’t confuse the coefficient of a radical with the index of
a radical.
3 3兹x ≠ 兹
x
Root of a Product
We have several rules of radicals, which are similar to the laws of exponents and, in fact, are
derived from them. The first rule is for products. By our definition of a radical,
兹n ab (ab)1/n
Using the law of exponents for a product and then returning to radical form,
(ab)1/n a1/nb1/n 兹 a 兹 b
n
n
So our first rule of radicals is as follows:
Root of a
Product
兹 ab 兹a 兹 b
n
n
n
38
The root of a product equals the product of the roots of the factors.
◆◆◆
Example 18: We may split the radical 兹 9x into two radicals, as follows:
兹 9x 兹 9 兹x 3兹x
since 兹 9 3.
◆◆◆
352
Chapter 13
◆◆◆
◆
Exponents and Radicals
Example 19: Write as a single radical 兹 7 兹 2 兹 x .
Solution: By Eq. 38,
兹 7 兹 2 兹 x 兹 7(2)x 兹 14x
There is no similar rule for the root of a sum.
Common
Error
◆◆◆
兹n a b ≠
兹a 兹b
n
n
Example 20: 兹9 兹 16 does not equal 兹 25 .
Common
Error
◆◆◆
◆◆◆
◆◆◆
Equation 31 does not hold when a and b are both negative
and the index is even.
Example 21:
(兹 4 )2 兹 4 兹 4 ≠ 兹 (4)(4)
≠ 兹 16 4
Instead, we convert to imaginary numbers, as we will show in Sec. 21–1.
◆◆◆
Root of a Quotient
Just as the root of a product can be split up into the roots of the factors, the root of a quotient
can be expressed as the root of the numerator divided by the root of the denominator. We first
write the quotient in exponential form,
a
pq
a
b
兹b
n
1/n
a1/n
b1/n
by the law of exponents for quotients. Returning to radical form, we have the following:
Root of a
Quotient
a
n
兹a
兹b
n
n 39
兹b
The root of a quotient equals the quotient of the roots of numerator and denominator.
◆◆◆
Rather than “simplifying,” we are
actually putting radicals into a
standard form, so that they may
be combined or compared.
Remember that we are doing
numerical problems as a way of
learning the rules. If you simply
want the decimal value of a
radical expression containing
only numbers, use your
calculator.
Example 22: The radical
w
兹w
兹w
can be written or .
兹 25
兹 25
◆◆◆
5
Simplest Form for a Radical
A radical is said to be in simplest form when:
1. The radicand has been reduced as much as possible.
2. There are no radicals in the denominator and no fractional radicands.
3. The index has been made as small as possible.
Removing Factors from the Radicand
Using the fact that
兹 x2 x,
兹3 x3 x, . . . ,
n n
兹
x x
try to factor the radicand so that one or more of the factors is a perfect nth power (where n is
the index of the radical). Then use Eq. 38 to split the radical into two or more radicals, some of
which can then be reduced.
Section 13–2
◆◆◆
◆
353
Simplification of Radicals
Example 23:
兹 50 兹 (25)(2) 兹 25 兹 2 5兹 2
◆◆◆
Example 24:
兹 x3 兹 x2x 兹 x2 兹 x x兹 x
◆◆◆
◆◆◆
◆◆◆
Example 25: Simplify 兹50x3 .
Solution: We factor the radicand so that some factors are perfect squares.
兹 50x3 兹 (25)(2)x2x
Then, by Eq. 38,
兹 25 兹 x2 兹2x 5x兹 2x
◆◆◆
◆◆◆
Example 26: Simplify 兹24y5 .
Solution: We look for factors of the radicand that are perfect squares.
兹24y5 兹4(6)y4y 2y2兹6y
◆◆◆
◆◆◆
Example 27:
兹24y 兹24y y y兹24y
4
◆◆◆
5
4
4
4
◆◆◆
3
Example 28: Simplify 兹
24y5.
Solution: We look for factors of the radicand that are perfect cubes.
3
24y5 兹8(3)y3y2
兹
3
3
3
3
兹
8y3 兹
3y2 2y 兹
3y2
◆◆◆
When the radicand contains more than one term, try to factor out a perfect nth power (where
n is the index).
◆◆◆
Example 29: Simplify 兹4x2y 12x4z .
Solution: We factor 4x2 from the radicand and then remove it from under the radical sign.
兹4x2y 12x4z 兹4x2(y 3x2z)
2x兹 y 3x2z
◆◆◆
Rationalizing the Denominator
An expression is considered in simpler form when its denominators contain no radicals. To put
it into this form is called rationalizing the denominator. We will show how to rationalize the
denominator when it is a square root, a cube root, or a root with any index and when it has more
than one term.
If the denominator is a square root, multiply numerator and denominator of the fraction
by a quantity that will make the radicand in the denominator a perfect square. Note that we are
eliminating radicals from the denominator and that the numerator may still contain radicals.
Further, even though we call this process simplifying, the resulting radical may look more complicated than the original.
354
Chapter 13
◆◆◆
◆
Exponents and Radicals
Example 30:
5
5
兹2
5兹 2
兹2
兹2
兹2
兹4
5兹2
2
◆◆◆
When the entire fraction is under the radical sign, we make the denominator of that fraction
a perfect square and remove it from the radical sign.
◆◆◆
Example 31:
3x
3x(2y)
6xy 兹6xy
◆◆◆
2y
2y(2y)
2y
4y2
If the denominator is a cube root, we must multiply numerator and denominator by a quantity that will make the quantity under the radical sign a perfect cube.
兹 兹
◆◆◆
兹
Example 32: Simplify
7
3
兹4
3
Solution: To rationalize the denominator 兹 4 , we multiply numerator and denominator of the
3 fraction by 兹
2 . We chose this as the factor because it results in a perfect cube (8) under the
radical sign.
3 3 3 7
7 兹2
7兹2
7兹2
◆◆◆
3 3 3 3 兹4
兹4 兹2
2
兹8
The same principle applies regardless of the index. In general, if the index is n, we must
make the quantity under the radical sign (in the denominator) a perfect nth power.
◆◆◆
Example 33:
5
2y
2y
兹x4
5
3兹
x
5
3兹
x
兹x4
5
5
2y 兹x4
2y兹x4
5
3x
5
3兹x5
◆◆◆
Sometimes the denominator will have more than one term.
◆◆◆
Example 34:
a
兹a b 兹
2
2
2
2
兹a(a2 b2)
a b
a
a2 b2 a2 b2
a2 b2
◆◆◆
Reducing the Index
We can sometimes reduce the index by writing the radical in exponential form and then reducing the fractional exponent, as in the next example.
◆◆◆
Example 35:
(a)
兹6 x3 x3/6
x1/2 兹x
Exercise 2
◆
4
4
(b) 兹
4x2y2 兹
(2xy)2
(2xy)2/4 (2xy)1/2
兹 2xy
Simplification of Radicals
Exponential and Radical Forms
Express in radical form.
1. a1/4
4. a1/2b1/4
x 1/3
7. p q
y
2. x1/2
5. (m n)1/2
8. a0b3/4
3. z3/4
6. (x2y)1/2
◆◆◆
◆
Section 13–2
355
Simplification of Radicals
Express in exponential form.
3
9. 兹 b
3 12. 4兹
xy
11. 兹 y2
n 14. 兹xm
10. 兹 x
n
13. 兹a b
16. 兹anb3n
n
15. 兹 x2y2
Simplifying Radicals
Write in simplest form.
17. 兹 18
18. 兹 75
19. 兹 63
3 21. 兹 56
23. 兹 a3
25. 兹 36x2y
20. 兹16
4
22. 兹48
5
24. 3兹
50x
3 2 5
26. 兹x y
4
3
32. x兹 x4 x3y2
34. 兹 2x3 x4y
33. 兹 9m3 18n
35. 兹x4 a2x3
36. x兹 a3 2a2b ab2
2
38.
5
2
40.
3
5
42.
8
7
44. 4 8
5m
46. 7n
5ab
48.
6xy
兹
兹
兹
兹
37. (a b)兹a3 2a2b ab2
兹
41.
兹
43.
兹
3
7
1
4
2
9
1
2x
3
3a
5b
3
3
45.
47.
兹
兹
兹
兹
49.
51.
兹
兹
3
6
1
2
x
4x6
9
50.
81x4
16yz2
x 2
兹x 2
p
q
2
3
pa q
a3 2
2
x
1
p
q
x
2
3
x
兹 x m y
兹
兹
54.
mn
兹
55. (m n)
兹
兹
3
52.
53. (x2 y2)
57.
30. 6兹16x4
31. 兹 a3 a2b
39.
28. 兹64m2n4
5
29. 3 兹
32xy11
3
3
3
16x3y
27. x兹
If you are using a calculator for
any of these problems, be sure
to leave the answers in radical
form.
56.
2
8a 48a 72
3a
Applications
58. The period n for simple harmonic motion is given by Eq. A37.
n Write this equation in exponential form.
兹
kg
W
For these more complicated
types, you should start by
simplifying the expression under
the radical sign.
356
◆
Chapter 13
Exponents and Radicals
59. The magnitude, Z, of the impedance of a series RLC circuit is given by Eq. A99.
Z 兹R2 X2
Write this equation in exponential form.
60. Given Eq. A99 from problem 59, write the expression for Z when X 2R, and
simplify.
61. The hypotenuse in right triangle ABC, shown in Fig. 13–2, is given by the
a
Pythagorean theorem.
B
c
c 兹 a2 b2
A
Write an expression for c when b 3a, and simplify.
62. In problem 58, rationalize the denominator in Eq. A37.
C
b
FIGURE 13–2
63. A stone is thrown upward with a horizontal velocity of 40 ft./s and an upward velocity
of 60 ft./s. At t seconds it will have a horizontal displacement H equal to 40t and a vertical displacement V equal to 60t 16t2. The straight-line distance S from the stone to the
launch point is found by the Pythagorean theorem. Write an equation for S in terms of t,
and simplify.
13–3
One reason we learned to put
radicals into a standard form is
to combine them.
Operations with Radicals
Adding and Subtracting Radicals
3
Radicals are called similar if they have the same index and the same radicand, such as 5 兹2x
3 and 3兹
2x . We add and subtract radicals by combining similar radicals.
◆◆◆
Example 36:
5兹
y 2兹 y 4兹 y 3兹 y
◆◆◆
Radicals that may look similar at first glance may not actually be similar.
◆◆◆
Example 37: The radicals
兹 2x and 兹 3x
are not similar.
◆◆◆
Common
Error
Do not try to combine radicals that are not similar.
兹 2x 兹 3x ≠ 兹 5x
Radicals that do not appear to be similar at first may turn out to be so after simplification.
◆◆◆
Example 38:
(a) 兹 18x 兹 8x 3兹 2x 2兹 2x 兹 2x
3
3
3
兹
81x3y 2y 兹
3y 3x 兹
3y
3
(2y 3x)兹3y
y
x
(c) 5y 2兹xy x y
x
xy
yx
5y 2 2兹 xy x 2
y
x
5兹 xy 2兹 xy 兹 xy 4兹 xy
(b)
兹24y
3
4
兹
兹
兹
兹
◆◆◆