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Radicals
Alex Moore
May 5, 2016
We now turn our attention to the study of radicals. We previously studied
basic use of square roots and will now pursue a more in-depth look, including
roots of degree other than 2. This has many reasons, chiefly among which
is the relationship between the degree of a function and the number of roots
(x-intercepts) it has.
1
The Basics
Before beginning new concepts, let’s review the basic rules of exponents and
square roots.
1.1
Exponents
Here’s a quick review of some elementary properties of exponents, which are
simply a shorthand way of writing repeated multiplication. For example,
21
22
23
24
25
=2
=2·2=4
=2·2·2=8
= 2 · 2 · 2 · 2 = 16
= 2 · 2 · 2 · 2 · 2 = 32
and so on.
1
Similarly, variables can be raised to exponents, such as
x1
x2
x3
x4
x5
1.2
=x
=x·x
=x·x·x
=x·x·x·x
=x·x·x·x·x
and so on.
Rules for Square Roots
This is just a quick review of some square root properties we learned in
previous sections. Nothing new here.
√
√ √
Product of Square Roots
a · b√= a · b
pa
a
√ =
Quotient of Square Roots
b
b
Recall that a square root is the inverse operation to squaring. In other
words, whereas squaring a number means taking that number (or variable)
and multiplying it by itself, a square root seeks to find which number satisfies
the squaring condition for the radicand (the number or expression under the
radical). See Alex’s Guide to Square Roots for further explanation.
Furthermore, note that it is not good mathematical practice to consider
something as “simplified” if there is a perfect square under a radicand, if a
radicand contains a fraction, or if there are any radicals in the denominator
of a fraction.
2
Radicals and Roots
What is the connection between radicals and roots?
2.1
nth Roots
The formal definition of the square root of a nonnegative number a is the
nonnegative number whose square equals a.1 The small number that indi√
(with no index)
cates the root is called the radical’s index. Note that
1
Israel M. Gelfand & Alexander Shen, Algebra, 1958.
2
√
and 2 mean the same thing. We can extend this line of thinking to cubes
and cube roots, and indeed, higher degrees as well. If you have a number
a, then the cube root of a is the number whose cube equals a. An example
would be
√
3
8=2
23 = 8 and
Besides “square” and “cube” there are no other fancy words for the degrees;
we simply say “fourth root,” etc. This can also be extended to the nth root
of a number.
2.2
The Connection with Exponents
Roots and radicals are directly related to exponents, as we saw before. Let’s
do a quick example to illustrate this connection.
Example. Prove that, for nonnegative a,
√
√ m
n
am = n a
Solution. We know that the radicand just means a multiplied by itself
m times. In other words, we can rewrite the left-hand side as
√
√
n
am = n a
| · a{z· · · a}
m times
Using the product of square roots rule, we have
√
√
√
n
a · n a··· n a
|
{z
}
m times
which equals
m
√
n
a
Remark. This relates to exponents in the definition of non-integer powers. What would a fraction exponent mean? Here we construct the definition
as
√
√
1
x /2 = 2 x = x
or
x /3 =
1
√
3
x ; x /4 =
√
4
x /n =
√
n
1
or in general
1
3
x ; x /5 =
1
x
√
5
x
This can also be extended to non-integer powers with something other
than 1 in the numerator. For example
√ 2 √
1
3
2
x /3 = x 3 ·2 = 3 x = x2
because of what we said in the example above. Try this out with some different numbers on the calculator and see what you think. This is an important
result to realize; radicals are separable, associative, and commutative.
√ 2
√ 10
Example. Show that ( 15 a) = ( 3 a) .
Solution. Separate each of the exponents into factors.
q 2
5
√ 2·5
5 √
3·5
3
a
=
a
Notice that the ( )5 and
√
5
cancel each other out, and we are left with
√ 2·5
a
=
3·5
3
q 2
5
√ 2
5 √
3a = 3a
Operations with Radicals
This section mostly consists of examples that illustrate how radicals behave
in actual problems.
3.1
Simplifying Radicals
We have been simplifying expressions all year, and now we will learn how to
simplify radicals. It may seem confusing, but all you need is a little patience
and ingenuity. The goal is to factor the radicand into squares (if dealing with
square roots) and pull them out by using the product of square roots rule.
Keep going until the only radicand left is not a perfect square. The same
logic also applies to radical indices other than 2.
√
Example. Simplify 90. Begin by separating the radicand into factors,
looking specifically for factors that are perfect squares.
√
90 =
√
10 · 9 =
√
4
3 √
√
>
10 · 9 = 3 10
Say, “Three square roots of 10.”
√
Example. Simplify 72. Follow the same line of thinking as the last
example. Factor the radicand and look for perfect squares.
√
72 =
√
2 · 36 =
√ √ *6 √
2 · 36 = 6 2
Say, “Six square roots of 2.”
3.2
Absolute Value
I want to briefly talk about the role of absolute value here. Let’s do a thought
experiment. We know that
42 = 4 · 4 = 16
which implies that
√
16 = 4. But if
(−4)2 = (−4) · (−4) = 16
√
then doesn’t it follow that 16 = −4? Both are correct, and we begin to see
the connection between radicals, exponents, and absolute value. The positive
root is called the principal root. This can be particularly tricky if the radicand contains variables. You must use absolute value to ensure nonnegative
results. When the radicand is negative, you have to use imaginary numbers,
which we will not cover in this class.
√
Example. Consider x2 which would appear to be the same as just x.
But what if x was a negative number, such as −2. Then
√
x2 = x ; x = −2
p
(−2)2 = −2
√
4 = −2
2 6= −2
and we see the issue with our thought process. To account for this and
prevent errors, we use the convention that
√
√
√
√
√
√
√
x2 = |x|
x3 = x x
x4 = x 2
x5 = x2 x
x6 = |x3 | and so on.
5
3.3
Like and Unlike Radicands
It logically follows, then, that the radicand determines how two radicals interact with each other.
√
√
√
Example. Simplify 6 5 + 2 5 − 5 5. Since all the radicands are the
same, we can treat
√ them like variables, and combine based on the “coefficients” to get 3 5.
√
√
√
Example. Simplify 6 27 + 8 12 + 2 75. Since the radicands are different, we must first factor them and look for perfect squares.
3
√
√
√
√
>√
6 27 can be rewritten as 6 9 · 3 = 6 9 3 = 18 3
2
√
√
√
√
>√
8 12 can be rewritten as 8 4 · 3 = 8 4 3 = 16 3
√
√
√ √
*√5
2 75 can be rewritten as 2 25 · 3 = 2
25 3 = 10 3
Therefore we can conclude that
√
√
√
√
√
√
6 27 + 8 12 + 2 75 = 18 3 + 16 3 + 10 3
√
= 44 3
√
√ √
√ Example. Simplify
5− 2
14 + 35 . We use the FOIL method
to make sure everything in the first group gets multiplied by everything in
the second group.
√ √
√
First would be 5 · 14 = 70
√ √
√
Outer would be 5 · 35 = 175
√ √
√
Inner would be − 2 · 14 = − 28
√ √
√
and Last would be − 2 · 35 = − 70
In this case, the first and last terms conveniently cancel, and we are left with
√
√
175 − 28
This can be simplified further, by following the same process of factoring the
radicands and looking for perfect squares.
√
25 · 7 −
2
√
√
√
√ * 5√
> √
4 · 7 = 25 · 7 − 4 · 7 = 3 7
6
3.4
Equations Involving Radicals
Perhaps the most indicative implication of these notes is how to deal with a
radical in an equation. This will be the most important thing to take away
from this section.
q
− 2 = 0. Begin to isolate the radical by moving
Example. Solve 5 4x
3
the −2 to the other side.
r
4x
5
=2
3
Divide through by the leading 5.
r
2
4x
=
3
5
Now square both sides.
4x
4
=
3
25
Cross-multiply the denominator of 3.
4x =
12
25
Finally, divide through by 4 to isolate x.
x=
12
100
Simplify the fraction to arrive at the solution.
x=
3
25
It’s a good idea to check your solution to make sure it works.
3.5
Extraneous Solutions
This is an unintended byproduct of solving equations with radicals. As we
saw in the preceding example, we squared the radical as a technique for isolating the variable when solving. The problem with doing that is that the
original equation wasn’t quadratic, and sometimes when you’re solving an
7
equation with radicals, you turn it into a quadratic without meaning to. To
compensate for the “extra solution” you get (because quadratics have two
solutions), you simply check what you get to make sure it satisfies the original equation. If a solution doesn’t work in the original equation, then we say
that solution is extraneous. Indeed, if the original equation has the variable
on both sides, one under a radical and one not, then you will unavoidably
change the original equation into a quadratic, and one of the solutions you
get will be extraneous. Here’s an example to show what this looks like.
√
Example. Solve 4 + x + 2 = x. We begin by isolating the radical by
moving the 4 to the other side.
√
x+2=x−4
Now square both sides. Notice that the x on the right-hand side will become
squared, and thus the equation becomes a quadratic.
x + 2 = (x − 4)2
Expand the notation on the right
x + 2 = (x − 4)(x − 4)
and FOIL the resulting binomials.
x + 2 = x2 − 8x + 16
Solve for 0 by moving the terms all to one side. For convenience, I will choose
the right-hand side.
0 = x2 − 9x + 14
Now factor the resulting polynomial. This is a quadratic trinomial with a = 1
so we need to look for factors that multiply to 14 and add to −9.
0 = (x − 7)(x − 2)
And thus we have two solutions
x=7
x=2
8
We need to check these to see which one is extraneous, since the original equation wasn’t quadratic. Plug each of the solutions into the original equation
and see what happens. For x = 7 we have
√
4+ 7+2=7
√
4+ 9=7
4+3=7
7=7
so this solution is good.
For x = 2 we have
4+
√
2+2=2
√
4+ 4=2
4+2=2
6 6= 2
so this solution is extraneous.
3.6
Rationalizing the Denominator
It is not proper mathematical convention to leave radicals in the denominator. This can easily be accomplished by multiplying the top and bottom
by the radical. The idea here is that when you multiply a radical by itself,
you’re squaring it, and the radical is cancelled out. This process is called
rationalizing the denominator.
2
Example. Simplify √ . This can be accomplished by multiplying by
3
the radical as a fraction over itself, which is an old algebraic trick.
√
√
√
√
2
3
2· 3
2· 3
2 3
√ · √ = √ √ = √ 2 =
3
3
3
3· 3
3
|{z}
=1
3.7
Conjugates
When you have other terms to worry about, dealing with a radical in the
denominator is often made easiest by utilizing its conjugate. This builds on
9
the concepts of the last section for rationalizing the denominator. Generally
speaking, the conjugate of a binomial is the same two terms with a different
sign in the middle.
2
√ . Since the denominator has more than one
6− 3
√
term in√it, we have to use its conjugate to rationalize. The conjugate of 6− 3
is 6 + 3. Use the same logic as the last section, but with the conjugate.
√
√
2(6 + 3)
2
2
6+ 3
√ =
√ ·
√ =
√
√
6− 3
6− 3 6+ 3
(6 − 3)(6 + 3)
| {z }
Example. Simplify
=1
Now notice that when the denominator is FOILed, the radicals will annihilate
each other by becoming squared.
√
√
√
√
12 + 2 3
12 + 2 3
12 + 2 3
2(6 + 3)
√
√ =
√
√
√
=
=
36 − 3
33
(6 − 3)(6 + 3)
36 + 6 3 − 6 3 − ( 3)2
10