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The Humongous Book of Algebra Problems by W. Michael Kelley

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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
youre 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.
Dont 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:
.

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