Chapter Thirteen — Radical Expressions and Equations
The Humongous Book of Algebra Problems
283
To simplify , divide the power of y by the index of the radical: 7 ÷ 4 = 1 with
remainder 3, so . Substitute
into the expression.
Group together the radicals of the expression.
13.20 Identify the value of x that makes the statement 16
x/4
= 8 true.
Rewrite the left side of the equation as a radical expression. In this case,
is preferred to .
The solution x answers the question, “Two raised to what power produces a
value of 8?” The answer is x = 3.
Radical Operations
Add, subtract, multiply, and divide roots
13.21 What condition must be met by radical expressions in order to calculate a sum
or difference?
Radical expressions can be added or subtracted only when they contain
“like radicals,” which means that the radicands and indices are equal. This
requirement is similar to the “like terms” requirement placed on polynomial
addition and subtraction, wherein terms are required to have equivalent
variable expressions before they can be combined.
13.22 Simplify the expression: .
Combine the coefficients of the like radicals to compute the sum:
2 – 5 + 14 = 11.
Solving the
“nonpreferred”
way to write the
radical equation
involves logarithms,
which aren’t covered
until Chapter 18. The
preferred version is
easier because x
isn’t inside the
radical.
“Indices” is
just the plural form
of “index.”
You’re
allowed
to add these
because they
all contain the
same radical
expression:
.
Chapter Thirteen — Radical Expressions and Equations
The Humongous Book of Algebra Problems
284
13.23 Simplify the expression: .
The radicals cannot be added because the radicands are not equal. However,
can be simplified.
The expression now consists of like radicals. Calculate the sum.
13.24 Simplify the expression: .
The cube roots contain factors that are perfect cubes; simplify both.
13.25 Simplify the expression: .
Apply the distributive property.
The product of two roots with the same index is equal to the root of the
product: .
Simplify the expression.
You can
subtract
the coefcients
because the radi-
cals match: 2 – 3 = –1.
Instead of writing a
coefcient of –1, just
use the negative
sign by itself.
Problem
13.5 said you
could split a
product inside
one radical into
separate radicals.
This problem tells you
it works backward
as well; two radicals
that are multiplied
can be merged into
one big radical
as long as they
have the same
index.
Chapter Thirteen — Radical Expressions and Equations
The Humongous Book of Algebra Problems
285
13.26 Simplify the expression: .
Express the squared quantity as a product.
Calculate the product using the technique described in Problem 11.19.
Combine like radicals.
Recall that .
13.27 Simplify the expression:
Apply the distributive property.
Combine like radicals and simplify.
13.28 Simplify the expression and express it using radical
notation.
According to a property of exponents, (x
a
)
b
= x
ab
.
Calculate the product.
If you don’t
understand why
you need absolute
values, look at
Problem 13.10.
When some-
thing raised to a
power (like x
1/2
or y
3/4
)
is raised to a power
(like 2), multiply the
powers together.
Chapter Thirteen — Radical Expressions and Equations
The Humongous Book of Algebra Problems
286
Write y
7/2
as a radical expression and simplify.
13.29 Calculate the quotient .
Express the quotient as a fraction.
The quotient of two roots with the same index is equal to the root of the
quotient.
Eliminate the negative exponent by moving x
–2
into the denominator.
Simplify the expression.
In other words,
instead of having
separate radicals
in the numerator and
denominator, put the
entire fraction under
one radical.
Chapter Thirteen — Radical Expressions and Equations
The Humongous Book of Algebra Problems
287
13.30 Rationalize the expression: .
A rationalized expression does not contain a root in the denominator of the
fraction. Rewrite the square root of this quotient as a quotient of square roots.
To eliminate in the denominator, multiply the numerator and denominator
by .
The square root is eliminated by replacing it with the perfect square 25.
13.31 Simplify the expression and rationalize the answer.
Write the quotient of cube roots as the cube root of a quotient and simplify the
fraction.
Eliminate the negative exponent by moving y
–2
into the denominator.
To rationalize the expression, the cube root in the denominator must be
eliminated. To accomplish this, multiply the numerator and denominator of
the fraction by .
Anything
divided by
itself equals 1
(including
divided by itself),
and multiplying by
1 doesn’t change
the fraction’s
value.
This fraction
is made up of
the coefcients in
the numerator and
denominator. There is
no explicit numerator up
top, so there’s an implied
coefcient of 1.
The denom-
inator starts
as 3y
2
. Multiplying
this by two more
3s, and one more y
gives you 3
3
y
3
. The goal
is to raise everything
inside the radical to a
power that matches
the index of the
radical, in this
case 3.
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