Section 5.1
Radical Notation and Rational Exponents 1
5.1 Radical Notation and Rational Exponents
We now review how exponents can be used to describe not only powers (such as 52 and
23 ), but also roots (such as square roots and cube roots). Along the way, we’ll define
higher roots and develop a few of their properties.
Integer Exponents
Recall that use of a positive integer exponent is simply a shorthand for repeated multiplication. For example,
23 = 2 · 2 · 2.
(5.1)
In general, bn stands for the quanitity b multiplied by itself n times. With this definition,
the following Laws of Exponents hold.
Laws of Exponents
1. br bs = br+s
2.
br
= br−s
bs
3. (br )s = brs
Recall that negative exponents, as well as the 0 exponent, are simply defined in
such a way that the Laws of Exponents will work for all integer exponents.
Negative exponents and the 0 exponent are defined as follows:
Definition 1
b−n =
1
bn
and
b0 = 1
provided that b 6= 0.
For example, 20 = 1, and 2−4 = 1/24 = 1/16.
We now have bn defined for all integers n, in such a way that the Laws of Exponents
hold. Recall that we can likewise define expressions using rational exponents, such as
21/3 , in a consistent manner. Before doing reviewing that material, we’ll need to take
a detour and review roots.
Roots
Square Roots: Let’s begin by reminding ourselves what the square root of a real
number represents.
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Chapter 5
Definition 2
Given a real number a, the “square root of a” is a number x such that x2 = a.
We investigate square roots in more detail by looking for solutions of the equation
x2 = a.
(5.2)
There are three cases, each depending on the value and sign of a.
Summary: Square Roots
The solutions of x2 = a are called “square roots of a.”
•
Case I: a < 0. The equation x2 = a has no real solutions.
Case II: a = 0. The equation x2 = a has one real solution, namely x = 0.
√
Thus, 0 = 0.
√
• Case III: √
a > 0. The equation x2 = a has two real solutions, x = ± a. The
square root of a, that is, the positive solution
notation a calls for the positive
√
of x2 = a. The notation − a calls for the negative square root of a, that is,
the negative solution of x2 = a.
•
Let’s look at some examples.
I Example 1
What are the solutions of x2 = −5?
We’re asked to find a solution of x2 = −5, so you must find a number whose
square equals −5. However, whenever you square a real number, the result is always
nonnegative
√ (zero or positive). It is not possible to square a real number and get −5.
That is, −5 is not a real number.
I Example 2
What are the solutions of x2 = 0?
There is only one solution, namely x = 0. Note that this means that
I Example 3
What are the solutions of x2 = 25?
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√
0 = 0.
Section 5.1
Radical Notation and Rational Exponents 3
√
The solutions of x2 = 25 are called square roots of 25 and are written x = ± 25.
In this case, we can simplify further and write x = ±5.
Cube Roots: Let’s move on to the definition of cube roots.
Definition 3
Given a real number a, a “cube root of a” is a number x such that x3 = a.
For example, 2 is a cube root of 8 since 23 = 8. Likewise, −4 is a cube root of
−64 since (−4)3 = −64. Thus, taking the cube root is the “opposite” of cubing, so the
definition of cube root must be closely connected to the graph of y = x3 , the cubing
function. Therefore, we look for solutions of
x3 = a.
(5.3)
A detailed summary of cube roots follows.
Summary: Cube Roots
The solutions of x3 = a are called the “cube roots of a.” Whether a is negative,
zero,√or positive makes no difference. There is exactly one real solution, namely
x = 3 a.
Let’s look at an example.
I Example 4
What are the solutions of x3 = −8?
√
The equation x3 = −8 has exactly one real solution, denoted x = 3 −8. Now since
(−2)3 = −8, it follows that x = −2 is a real solution of x3 = −8. Consequently, the
cube root of −8 is −2, and we write
√
3
−8 = −2.
√
Again, because there is only one real solution of x3 = −8, the notation 3 −8 is pronounced “the cube root of −8.” Note that, unlike the square root of a negative number,
the cube root of a negative number is allowed.
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Chapter 5
Higher Roots: The previous discussions generalize easily to higher roots, such as
fourth roots, fifth roots, sixth roots, etc. The key is to remember that all even roots
behave the same as the square root and all odd roots behave the same as the cube root.
Summary: Even nth Roots
If n is a positive even integer, then the solutions of xn = a are called “nth roots
of a.”
Case I: a < 0. The equation xn = a has no real solutions.
Case II: a = 0. The equation xn = a has exactly one real solution, namely
√
x = 0. Thus, n 0 = 0.
√
• Case III: √
a > 0. The equation xn = a has two real solutions, x = ± n a. The
notation n a calls for the positive
nth root of a, that is, the positive solution
√
of xn = a. The notation − n a calls for the negative nth root of a, that is, the
negative solution of xn = a.
•
•
Summary: Odd nth Roots
If n is a positive odd integer, then the solutions of xn = a are called the “nth
roots of a.” Whether a is negative, zero, or positive
makes no difference. There is
√
n
n
exactly one real solution of x = a, denoted x = a.
Rational Exponents
Recall that rational exponents are defined in such a way that the Laws of Exponents
still apply. For example, if n is any odd positive integer, then 21/n must be defined by
the formula
√
1
n
2 n = 2.
With this definition, the Laws of Exponents hold for all rational exponents.
Definition 4
For a positive rational exponent
m
bn
m
n,
and b > 0,
√
√
n
n
= bm = ( b)m .
(5.4)
For a negative rational exponent − m
n,
m
b− n =
1
m .
bn
(5.5)
1
For b < 0, the same definitions make sense only when n is odd. For example (−2) 4
is not defined.
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Radical Notation and Rational Exponents 5
Section 5.1
Remark 1. For b < 0, the same definitions make sense only when n is odd. For
1
example (−2) 4 is not defined.
I Example 5
Simplify the following expressions, and write them in the form xr :
2
2 1
2
1
x3
4
b) 1 ,
a) x 3 x 4 ,
c) x− 3
4
x
2
1
2
8
1
11
3
a) x 3 x 4 = x 3 + 4 = x 12 + 12 = x 12
2
b)
x3
1
4
2
1
8
3
5
= x 3 − 4 = x 12 − 12 = x 12
x
2 1
2 1
2
1
4
= x− 3 · 4 = x− 12 = x− 6
c) x− 3
Multiplication Properties of Radicals
Recall the following property of radicals.
Property 1
Let a and b be positive real numbers. Then,
√
√ √
ab = a b.
(5.6)
This result can be used in two distinctly different ways.
•
You can use the result to multiply two square roots, as in
√ √
√
7 5 = 35.
•
You can also use the result to factor, as in
√
√ √
35 = 5 7.
One question that arises is when is a radical in reduced form? We use the following
set of guidelines for a special form of the answer which we will call simple radical form.
The First Guideline for Simple Radical Form. When possible, factor out a
perfect square.
√
For example, 32 is not in simple radical form, as it is possible to factor out a
perfect square, as in
√
√ √
√
32 = 16 2 = 4 2.
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Chapter 5
Let’s place another radical expression in simple radical form.
I Example 6
√
Place 50 in simple radical form.
Because 50 = 25 · 2, we can use Property 1 to write
√
√ √
√
50 = 25 2 = 5 2.
When working with square roots it can be extremely useful to recall the definition
of a rational exponent. The process of placing a square root into simple radical form
can be done much more efficiently using rational exponents.
Taking the Square Root of an Even Power. When taking a square root of
xn , when x is a positive real number and n is an even natural number, divide the
exponent by two. In symbols,
√
xn = xn/2 .
Also, recall that raising a product to a power requires that we raise each factor to
that power.
Raising a Product to a Power.
(ab)n = an bn .
Let’s look at an example that employs these properties.
I Example 7
√
Simplify 25 33 .
In this example, the difficulty is the fact that the exponents are not divisible by 2.
However, if possible, the “first guideline of simple radical form” requires that we factor
out a perfect square. So, extract each factor raised to the highest possible power that
is divisible by 2, as in
√
√
√
25 33 = 24 32 2 · 3
Now, divide each exponent by 2.
√
√
√
24 32 2 · 3 = 22 31 2 · 3
Finally, simplify by expanding each exponential factor and multiplying.
√
√
√
22 31 2 · 3 = 4 · 3 2 · 3 = 12 6
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Section 5.1
Radical Notation and Rational Exponents 7
When there are variables in the radicand,
then we need to be careful how we simplify.
p
Note that ifpwe incorrectly reduce (−3)2 , we might be tempted to say it is −3. But
recall that (−3)2 is a positive square root and the resulting simplification also needs
to be positive. This discussion leads to the following result.
The Positive Square Root of the Square of x. If x is any real number, then
√
x2 = |x|.
Let’s use these ideas to simplify some radical expressions that contain variables.
Variable Expressions
I Example 8
Given that the x represents any real numbers, place the radical expression
√
48x6
in simple radical form.
Simple radical form demands that we factor out a perfect square, if possible. In
this case, 48 = 16 · 3 and we factor out the highest power of x that is divisible by 2.
√
√
√
48x6 = 16x6 3
We can now use Property 1 to take the square root of each factor.
√
√
√ √ √
16x6 3 = 16 x6 3
√
calls for a nonnegative square root, so we must
Now, remember that the notation
insure that each response in the equation above is nonnegative. Thus,
√
√ √ √
16 x6 3 = 4|x3 | 3.
Division Properties of Radicals
We continue our review by stating the following property of radicals.
Property 2
Let a and b be positive real numbers. Then,
r
√
a
a
= √ .
b
b
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Chapter 5
Simple Radical Form Continued
With this property we can add two more guidelines for simple radical form.
Simple Radical Form. When your answer is a radical expression:
1. If possible, factor out a perfect square.
2. Don’t leave fractions under a radical.
3. Don’t leave radicals in the denominator of a fraction.
If x is any real number, recall again that
√
x2 = |x|.
Let’s look at another example.
I Example 9
p
18/x6 in simple radical form. Discuss the domain.
p
Note that x cannot equal zero, otherwise the denominator of 18/x6 would be zero,
which is not allowed. However, whether x is positive or negative, x6 will be a positive
number (raising
p a nonzero number to an even power always produces a positive real
number), and 18/x6 is well-defined.
Place the expression
Keeping in mind that x is nonzero, but could either be positive or negative, we proceed by first invoking Property 2, taking the positive square root of both numerator
and denominator of our radical expression.
r
√
18
18
=√
6
x
x6
From the numerator, we factor a perfect square.
value bars to insure a positive square root.
√
√ √
18
9 2
√ =
=
|x3 |
x6
In the denominator, we use absolute
√
3 2
|x3 |
We can use the Product Rule for Absolute Value to write |x3 | = |x2 ||x| = x2 |x|. Note
that we do not need to wrap x2 in absolute value bars because x2 is already positive.
√
√
3 2
3 2
= 2
|x3 |
x |x|
Because x could be positive or negative, we cannot remove the absolute value bars
around x. We are done.
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Section 5.1
Radical Notation and Rational Exponents 9
Radical Expressions
Now that we have reviewed how to multiply and divide square roots, we will simplify
a number of more extensive expressions containing square roots,
I Example 10
√
√
Simplify the expression (2 12)(3 3). Place your answer in simple radical form.
√
√
We’ll first take the product of 2 and 3, then the product of 12 and 3, then
multiply these results together.
√
√
√ √
√
(2 12)(3 3) = (2 · 3)( 12 3) = 6 36
√
Of course, 36 = 6, so we can simplify further.
√
6 36 = 6 · 6 = 36
Recall the following operation, where you “distribute the 2,” multiplying each term
in the parentheses by 2.
2(3 + x) = 6 + 2x
You can do precisely the same thing with radical expressions.
√
√
2(3 + 5) = 6 + 2 5
Like the familiar example above, we “distributed the 2,” multiplying each term in the
parentheses by 2.
Let’s look at an example.
I Example 11
√
√
Use the distributive property to expand the expression 12(3 + 3), placing your final
answer in simple radical form.
√
√
First,
distribute
the
12,
multiplying
each
term
in
the
parentheses
by
12. Note
√ √
√
that 12 3 = 36.
√
√
√
√
√
12(3 + 3) = 3 12 + 36 = 3 12 + 6
However, this last expression is not in simple radical form, as we can factor out a perfect
square (12 = 4 · 3).
√
√ √
3 12 + 6 = 3( 4 3) + 6
√
= 3(2 3) + 6
√
=6 3+6
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Chapter 5
The distributive property is also responsible in helping us combine “like terms.”
For example we know that 3x + 5x = 8x. It is the distributive property that actually
provides this solution. Note how we use the distributive property to factor x from each
term.
3x + 5x = (3 + 5)x
Hence, 3x + 5x = 8x. You can do the same thing with radical expressions.
√
√
√
3 2 + 5 2 = (3 + 5) 2
√
√
√
Hence, 3 2 + 5 2 = 8 2, and the structure of this result is identical to that shown
in 3x + 5x = 8x. There is no difference in the way we combine these “like terms.” We
repeat the common factor and add coefficients. In the case that we don’t have like
terms, as in 3x + 5y, there is nothing to be done. In like manner, each of the following
expressions have no like terms that you can combine. They are as simplified as they
are going to get.
√
√
√
√
√
√
3 2 + 5 3, 2 11 − 8 10, and 2 x + 5 y
However, there are times when it can look as if you don’t have like terms, but when
you place everything in simple radical form, you discover that you do have like terms
that can be combined by adding coefficients.
I Example 12
√
√
Simplify the expression 5 27+8 3, placing the final expression in simple radical form.
We can extract a perfect square (27 = 9 · 3).
√
√
√ √
√
5 27 + 8 3 = 5( 9 3) + 8 3
√
√
= 5(3 3) + 8 3
√
√
= 15 3 + 8 3
Note that we now have “like terms” that can be combined by adding coefficients.
√
√
√
15 3 + 8 3 = 23 3
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