Chapter Thirteen — Radical Expressions and Equations
The Humongous Book of Algebra Problems
276
Simplifying Radical Expressions
Moving things out from under the radical
Note: Problems 13.1–13.2 refer to the radical expression .
13.1 Identify the radicand and index of the expression.
The index of a radical expression is the small number outside and to the left of
the radical symbol; the index of is 3. The value inside the radical symbol is
called the radicand; the radicand of is 64.
Note: Problems 13.1–13.2 refer to the radical expression .
13.2 Simplify the expression and verify your answer.
As explained in Problem 13.1, the index of is 3. To simplify the cube root,
try to identify factors of the radicand that are perfect cubes. It is helpful to
memorize the first six perfect cubes, listed here.
Notice that 64 is a perfect cube, as 4
3
= 64. Rewrite the radical expression,
identifying the perfect cube explicitly.
Any factor of the radicand that is raised to an exponent equal to the index of
the radical can be removed from the radical.
Therefore, . To verify this solution, raise the answer (4) to an exponent
equal to the index of the original radical (3). The result should be the original
radicand (64).
4
3
= 64
Note: Problems 13.3–13.4 refer to the radical expression .
13.3 Identify the index and radicand of the expression.
The radicand of is the number inside the radical symbol: 36. No index is
written explicitly, which indicates an implied index of 2. Square roots are rarely
written with an explicit index, so the notation is preferred to .
Radicals
with index
3 are called
“cube roots,”
just like raising
something to the
third power is called
“cubing” it. Radicals
with index 2 are
called “square
roots,” just like
raising something
to the second
power is called
“squaring” it.
The 4
inside the
radical symbol
has exponent 3,
and the index of
the radical (that
tiny number oating
out front) is also 3.
When the power and
the index match, they
sort of cancel each
other out. The 3
exponent goes away,
the index goes
away, and the
radical sign
disappears,
too.
When there’s no little
number nestled in the check mark
outside the radical, that means
you’re dealing with a square root,
and you can assume the index
is 2.
Chapter Thirteen — Radical Expressions and Equations
The Humongous Book of Algebra Problems
277
Note: Problems 13.3–13.4 refer to the radical expression .
13.4 Simplify the expression and verify your answer.
According to Problem 13.3, the index of is 2. To simplify the square root,
factor the radicand using perfect squares. It is helpful to memorize the first 15
perfect squares, listed here.
Notice that 36 is a perfect square, as 6
2
= 36. Rewrite the radical expression,
identifying the perfect square explicitly.
As stated in Problem 13.2, any factor in the radicand that is raised to a power
equal to the index of the radical can be simplified.
To verify that , raise the answer (6) to a power that equals the index of
the original radical (2). The answer must match the original radicand (36).
6
2
= 36
13.5 Simplify the radical expression: .
The index of is not written explicitly, so according to Problem 13.3, the
index is 2 and the radical expression is a square root. Consider the perfect
squares listed in Problem 13.4 and determine whether any of those values divide
evenly into the radicand.
Because dividing 50 by 25 produces no remainder (50 ÷ 25 = 2), 25 is a factor of
50. Rewrite the radicand in factored form, explicitly identifying 25 as a perfect
square.
The root of a product is equal to the product of the roots.
The exponent of 5 and the index of the radical expression are equal, so simplify
the radical expression: .
Therefore, .
The index of
the radical is 2,
and the exponent
of 6 is 2. Those 2’s
cancel out and take
the radical sign
away with them. All
that’s left is 6.
Don’t even
bother with
1
2
= 1. Sure 1 is a
perfect square, but
it divides into every
number evenly and is
not all that useful
for simplifying
radicals.
In other words,
when two things are
multiplied inside a
root (a radical symbol),
you can break them
into two separate roots
that are multiplied:
. This
does NOT necessarily
work when two things
inside radicals are
added or subtracted:
.
Chapter Thirteen — Radical Expressions and Equations
The Humongous Book of Algebra Problems
278
13.6 Simplify the radical expression: .
The index of the radical expression is 3, so to simplify the expression, you must
identify factors that are perfect cubes. Consider the abbreviated list provided
in Problem 13.2. Notice that 64 divides evenly into 256: 256 ÷ 64 = 4. Factor 256
and identify the perfect cube explicitly.
The root of a product is equal to the product of the roots.
Simplify , noting that the exponent of the radicand is equal to the index of
the radical: .
13.7 Simplify the radical expression: .
The index of the radical is 5, so simplifying the expression requires you to
identify factors with an exponent of 5. Notice that 2
5
= 32, which divides evenly
into the radicand: 96 ÷ 32 = 3. Rewrite the expression, explicitly identifying the
factor that is raised to the fifth power.
13.8 Simplify the radical expression: .
The index of the radical is 3, so to simplify the expression, express x
6
as a
quantity raised to the third power. To accomplish this, divide the exponent of x
(6) by the index of the radical (3): 6 ÷ 3 = 2, so (x
2
)
3
= x
6
.
Now that the index of the radical and the exponent of the radicand are equal,
simplify the expression.
Don’t bother
ipping back to
Problem 13.2. Here’s
the list: 2
3
= 8, 3
3
= 27,
4
3
= 64, 5
3
= 125,
6
3
= 216.
There’s
no way that
3
5
= 243 could
be a factor of 96
because 243 is bigger
than 96. The same
goes for any number
larger than 3 raised
to the fth power, so
don’t even bother
checking it.
If you have a
hard time turning x
6
into
(x
2
)
3
to get the exponent of 3
you need, break x
6
into as many x
3
’s
as you can: x
6
= x
3
· x
3
(since x
3
· x
3
=
x
3 + 3
= x
6
). Now simplify:
Chapter Thirteen — Radical Expressions and Equations
The Humongous Book of Algebra Problems
279
13.9 Simplify the expression: .
To rewrite the radicand to include perfect cubes, divide each of the variables’
powers by 3, noting the resulting quotients and remainders. Begin with
x
17
: 17 ÷ 3 = 5 with remainder 2. Therefore, x
17
= (x
5
)
3
· x
2
.
Apply the same procedure to y
4
: 4 ÷ 3 = 1 with remainder 1, so y
4
= (y
3
)
1
· y
1
.
Finally, express z
8
using perfect cubes: 8 ÷ 3 = 2 with remainder 2, so
z
8
= (z
2
)
3
· z
2
. Rewrite the radicand using the above factorizations.
Separate the product into two radicals, one that contains factors raised to the
third power and one that contains factors raised to other powers.
Simplify the radical containing the perfect cubes.
13.10 Simplify the expression: .
The exponent of the radicand (2) is equal to the index of the radical (2) so the
expression can be simplified: . Note the presence of absolute values,
which were not required in the preceding problems of this chapter.
If n is an even number, then . In other words, if a factor in the
radicand is raised to an even power that is equal to the index of the radicand,
when simplified, that variable expression must appear in absolute values.
According to mathematical convention, a root with an even index must produce
a positive number. This holds true for natural numbers as well. Whereas this can
easily be ensured for natural numbers, absolute values are used to ensure that
roots of variables are positive.
13.11 Simplify the expression: .
Express the coefficient as a product that includes a perfect square:
. Express x
9
using a perfect square by dividing its
exponent by the index of the radical: 9 ÷ 2 = 4 with remainder 1, so x
9
= (x
4
)
2
·
x
1
. Apply the same procedure to y
3
: 3 ÷ 2 = 1 with remainder 1, so y
3
= (y
1
)
2
· y
1
.
Separate the radicand into two radicals, one that contains all of the perfect
squares and another that contains the remaining factors.
To rewrite
x
17
as something
to the third power,
you divide 17 by 3. It
goes in 5 times evenly:
(x
5
)
3
= x
15
. You still have
a remainder of 2, so
multiply by x raised to
the remainder power:
(x
5
)
3
· x
2
= x
17
.
RULE OF
THUMB: If a
radical has an
even index, then
simplifying it has
to result in positive
values. Even though 2
2
and (–2)
2
both equal
4,
is correct,
and
. You
don’t know if x is
positive or negative in
Problem 13.10, so use
absolute values to
make sure that
the answer’s
positive.
Chapter Thirteen — Radical Expressions and Equations
The Humongous Book of Algebra Problems
280
Simplify the radical containing the perfect squares. Note that and
because x
4
and y are raised to an even power that matches the index of
the radical.
13.12 Simplify the expression: .
To simplify the coefficient, identify factors that are raised to the fourth power:
. Divide the exponents of the variables by the index
of the radical to express the variables as factors raised to the fourth power:
7 ÷ 4 = 1 with remainder 3, so x
7
= (x
1
)
4
· x
3
; and 12 ÷ 4 = 3, so y
12
= (y
3
)
4
. The
factor z
3
cannot be rewritten because its power is less than the index.
Wondering
why x
4
suddenly jumped
out of those
absolute value bars?
x is raised to an even
power, so x
4
will always
be positive, no matter
what x equals.
However, y is not
raised to an even
power, so it has
to stay inside
the absolute
values.
Try dividing
405 by a few
natural numbers
raised to the fourth
power: 2
4
= 16, 3
4
= 81,
4
4
= 256, 5
4
= 625. Of
those, only 81 divides
evenly: 405 ÷ 81 = 5
so 3
4
· 5 = 405.
Put x and
y
3
in absolute
values because
they’re raised to
even powers that
match the index of
the radical. You can’t
drop those absolute
value bars because
x
1
and y
3
have odd
powers and could
be negative.
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