The
symbol √ is
called the radical sign. It means root.
Example:


square root, the 2 is usually
not written, it is implied


cube root


fourth root


fifth root



The number on
the radical sign is called the index. It specifies the root.
√ acts as
a grouping symbol.
√ 4 means
the square root of 4.
√ 4 +
6 means the square root of four first, then
add that to 6.
√ (4
+ 6) means the square root of the sum of 4 and
6, that is, the square root of 10.
We can use a vinculum above the sum
and remove the left and right parens like so:
It is common to say "the sum of 4
and 6 is under the
radical sign."
The expression
under the radical sign is called the
radicand.
In the
previous example (4 + 6) is the radicand.
At this point it will help to memorize
the
power table.
A root is a number
which multiplied by itself index number of times gives
the radicand as the result.
Example: 2 • 2
= 4, so 2 is the square root of 4 which is the radicand of √ 4 . (index
is 2)
Example: 3 is the
cube root of 27 since 3 • 3 • 3 = 27 which is the radicand
of . (index is 3)
You will notice that (2)
• (2) = 4 satisfies the definition of a root. 2 is
a square root of 4 as well.
The positive root is called
the principle root. By convention, the radical sign by itself
means the principle root.
If both roots are required
then the plusminus sign, ±, must be used.
Example: ±√ 16 is 4 and
4, which is writtem ±4
Even
roots of a negative number will not be allowed for the Real domain. This
restriction will be removed when we consider the Complex Number domain.
Odd roots of negative numbers are fine.
√ (4) is
not allowed. But is fine. 2 is the cube root of 8.
(2)(2)(2) = (4)(2) = 8
Example: is ± (2) which is 2 and 2. 2 is
not a root here
since 2
• 2 • 2 • 2 • 2 = 32 not the radicand 32.
So the root here is just
2.
The radical sign is a
fractional exponent. Example, √
4 is 4^{½}
and is 8^{1/3}
Notice how the index becomes the
denominator of the exponent. This will become useful
later.
Since radicals are exponents then
radicals obey the rules for
exponents .
Simplifying and
Evaluating Radicals
Multiplying radicals
Recall the
power rule of exponents
across factors.
Except when symplifying radicals we move from the right side of the
equation to the left like so:
For
example:
using the product rule, a = 12, b = 3 and n =
½
Usually we are multiplying radicals with the same indices.
But this does not need to be the case. When they are different
we need to first rewrite the smaller index in terms of the larger
index.
Example:
Note that 5 ^{½} = 5
^{¼} ^{+
¼} = 5 ^{¼}5^{¼}
so we can rewrite as
and we have which is
and finally:
Radicals with different indices with no common
fators can be multiplied but you need to find the least common multiple of the indices involved.
Example:
We need to the write the indices 2 and 3 in terms of 6.
Recalling that these indices are fractions we can proceed like so:
Radicals with fractional
indices
Recall the
power rule for exponents:
Example:
(16^{2})^{4} = 16^{8}
Example:
(16^{4})^{2} = 16^{8}
The index of the radical is by definition the fractional exponent of the radicand. The numerator of the fraction is the power of
the radicand; the denominator of the fraction is as usual the index of the radical.
Example: √x =
x^{½}
So
√x^{3} means
(x
^{3}
)^{½}
and using the power rule for exponents this
becomes
x^{3/2}
Moving the
other way,
Example:
In other
words, the square root of a quantity squared is that quantity.
This is true for any root.
Example: (we'll use the technique in this example in
what follows)
another way
to look at this is
Study the following carefully. The cummative law of multiplication is used on the exponents
to write the exponential in two equivalent radical forms.
To summarize,
the root of a power is the power of the root.
Moving factors to the outside
of the radical
Factors of the radicand that have
a power that is a multiple of the index can be brought outside the
radical.
Example: √16 = √(4^{2}) = 4
Note: familiarize yourself with the radical sign as used
here in its abbreviated form.
Example:
Example:
Here we need 16 in terms of a
power of the index 3. 2^{4} will not work. But we notice that
2^{4} can be written as (2^{3})• 2 and we have the
index matching the power of the factor (2^{3}).
Example:
The prime factors of 216 are helpful
here. Let's use repeated division.
216/2 = 108
108/2 = 54
54/2 = 27
and 27 = 3^{3}
So we must have
Of course, knowing that
6^{3} = 216 makes this a bit faster.
Moving factors to the inside
of the radical.
Here we proceed in reverse.
Simply raise the factor to the index and write it inside the
radical.
Example:
The index is 2, so we raise 4 to
the second power and write it inside like so:
Example:
We have:
Dividing
Radicals
We use the same rule for exponents as we did for
multiplication noting the following:
Example:
The indices
are the same, so we can combine under one radical
sign.
Example:
Examples with different indices:
Strategy: rewrite radical factors so they have the
same index so they can be combined then simplified.

The indices are 2 and 4.
8^{½} = 8^{¼}
^{+} ^{¼} = 8^{¼}
8^{¼}
Now all
radicals have the same index.
Combine under one radical sign.
(division rule)
8/2 = 4
32 = 2^{5} = 2^{4}
2

Radicals with algebraic expressions.
We assume that the radicand is positive for even indices.
We are not allowing the square root of a negative
number. For odd indices the radicand can be either positive or
negative.
Example: is ok, but is not ok
is ok, as is
When algebraic expressions are the radicands then we limit
the range of the independent variable in that expression to
make sure the radicands are valid.
Examples:

x can not be negative here, x
≥
0


x can be positive, negative, or zero
since its square is positive (or zero).


x can not be negative here, x
≥
0; if x were negative then
x^{3}
is less than 0 and we'd have the fourth root of a negative number
which we are not allowing.


x can be be positive, negative, or
zero


x can be zero or negative but not
positive. If x were positive then x is negative and the
radicand would be negative.


The radicand must be greater than or
equal to zero, so
x – 10 ≥ 0, which limits x
≥
10

Simplifying
Radical expressions
Simplifying radical expressions means to move all possible
factors outside the radicands and simplifying the result so that only one radical
with extermal factors remains. Note, when simplified, a radical may not exist.
Also,
if radicals exist in denominators they must be moved to
the numerator by a method called rationalizing the
denominator.
Examples:

Here the radicand has a power higher
than the index. So we must move the corresponding factor
out.
The index is 2 so we must remove the
highest power of each factor that has a power that is a multiple of 2.
Here that factor is x^{2}.
We can simplify no further.



The highest power of x that is a
multiple of the index 2 is 4.
So we factor the radicand into
x^{4}
x
Next we write x^{4} in terms
of the index (red 2).
Now we pull out the factor x^{2}.
There is nothing more we can
do.



Here's another example which is the
same as the previous
but with a larger exponent on the
radicand. Notice that the factor pulled out is a multiple
of the index. That multiple is brought outside the
radical. (The original power divided by the
index.)
Example if the index were 4 and the
factor had the power 12 then the resulting exponent of
the factor pulled out would be 12/4 = 3.



Here we note that 16 =
2^{4}, but 4 is not a multiple of 3 but if we write this
factor as 2^{3}2^{2} then we have
a factor with power 3, likewise for x^{5}, which
can be written as x^{3}x^{2}.
So we rewrite the radicand in terms of these factors.
and bring out the cubed factors.



Already simplified we have two terms,
we need one. Note, this can be factored to give us (x+2)(x2) to give us one term with two factors,
but these factors are different.



Here we have one term with two factors
5 and x^{2}. We can pull out the factor
x^{2}.



Here we have two terms but we can
factor the 16 from these terms to get 16(x – 2), a
term with two factors.
Now we have one term with a factor that
is the 4^{th} power of 2.
We can pull out the 2.



Our first step is to convert these
three terms into one term. We recognize the square of the
difference pattern and rewrite the radicand.
Now that we have a single term, we pull
out the factor (x – 2).
Notice,we do not need to constrain x
since (x – 2)^{2} is always non
negative.



Here we see x^{3} can be
written as x^{2}x, y^{4} as
(y^{2})^{2}
Write the single radical as the quotient of two
radicals.
Bring out the factors and simplify.
x cannot be negative, that is x
≥
0; y can be any value.

Rationalizing the denominator
is not simplified. We need
to remove the radical from the denominator. To do so we
note that and multiply by this fraction.
Now recall that , so the denominator becomes 2 and the
numerator becomes and we have:
Example:
We can write this as:
So we have
Now, the denominator names a cube root. That means we need
to multiply this cube root by itself 3 times to remove the radical, so we need
2 more factors. The fraction we need is:
Now multiply
= =
Many times we
need to use the special binomial product called the sum and
difference to rationalize the denominator.
Recall that: (a + b)(a – b) = a^{2} 
b^{2}
So if we have this:
then to remove the radical we can multiply by
Now
So we have =
Example:

Rearrange
the numerator to get the terms in the same order before
multiplying.

We could have saved a bit of work if we had first noticed
that the numerator is the negative of the denominator. Factor
out 1 from either then cancel.
Like so:



This goes to show that it never hurts
to take a second look before diving into the
problem.

Example:
As usual we simplify first then try combining like
terms.
Since 4 = 2^{2}, we can
bring the 2 outside.
Now is common to both terms so we can factor it
out to get
.
Solving Radical
Equations
goal: Eliminate radicals from the equation to allow
solving the equations by normal means.
strategy: Isolate one radical onto one side of the
equation to raise the equation to the index of that radical so
that the radical is eliminated and the equation can then be
solved.
method: Manipulate the equation using
addition/subtraction and multiplication/division to isolate the
radical. Raise the equation to the power of the index. Repeat
if multiple radicals exist in the equation until all radicals are
eliminated.
check: Verify the answers in the original
equation.
First we need the Principle of Powers:
For any positive integer n, if a = b then
a^{n}=
b^{n}.
This principle allows us to "raise an equation" to any power.
However, by doing so we can introduce solutions that
may not solve the original equation. We will see this effect as we proceed.
Example:
Here we have a single radical that is already isolated. The
index is 2 so we square both sides of the equation.
x  4 =
25
x = 29
check:
Example:
The radical is isolated and the
index is 3. We raise the equation to the 3^{rd}
power, that is, we cube the equation.

2x – 3 =
27
2x = 24
x = 12


check:
Example:

add x to both sides to isolate the radical
now square both sides (index is 2) to eliminate the radical
move all terms to one side and simplify the equation
solve the resulting quadratic equation
we find that x can be 6 or 1
x = 6 does not solve the original equation.
x = 1 is the solution.

In the previous example we obtained
two solutions, one of which was extraneous, x = 6.
Example:
This example has no Real
solution. The index is even which means the root must be
positive. Here the root is negative, that is, it is
6.
Example:
At least one radical is isolated,
so
can square both sides of the equation. We'll find that we'll still have the other radical, but
we can isolate it then square the equation again to eliminate it, like so:
Now square both
sides again.
m^{2} 28m + 196 = 16(m +
7)
m^{2}– 28m + 196 = 16m +
112
m^{2}  44m + 84 =
0
(m – 42)(m – 2) =
0
m = 42, m =
2
Check:
m = 42 is an
extraneous root.
m = 1 is the
solution to this equation.
Example:
First note that
x cannot equal 2 since we cannot divide the 4 by
zero.
Now
clear the denominator by multiplying the equation
by .
Now isolate the
radical.
The index is 2,
so square both sides. and solve the resulting quadratic
equation.
9(x + 2) = 4 – 4x +
x^{2}
9x + 18 = 4  4x +
x^{2}
0 = 14 – 13x +
x^{2}
x^{2}  13x 14 =
0
(x – 14) (x + 1) =
0
x = 14, x =
1
check:
Evaluate each
side of the equation and check if they equal each
other.
x =1 is a
solution.
14 is an
extraneous root.
x = 1 is the
solution to this equation.
Example:
The radicals
are isolated, but if we square both sides of the equation we're
left with a radical on the right hand side. If we raise the
equation to the 4^{th} power then both radicals
are eliminated.
We get x = 0,
and x = 7/9 as possible answers
Checking x = 0,
we find that the square root of 1 equals the fourth root of
1.
Checking 7/9,
in the left radical we get the square root of a negative
number,
so this is not
a solution. (3(7/9) + 1) = 7/3 + 1 = 4/3 < 0). 7/9 is an
extraneous root.
The solution to
this equation is 0.
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