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Section 1.1—Radical Expressions:
Rationalizing Denominators
Now that we have reviewed some concepts that will be needed before beginning
the introduction to calculus, we have to consider simplifying expressions with
radicals in the denominator of radical expressions. Recall that a rational number
is a number that can be expressed as a fraction (quotient) containing integers.
So the process of changing a denominator from a radical (square root) to a rational
number (integer) is called rationalizing the denominator. The reason that we
rationalize denominators is that dividing by an integer is preferable to dividing
by a radical number.
In certain situations, it is useful to rationalize the numerator. Practice with
rationalizing the denominator prepares you for rationalizing the numerator.
There are two situations that we need to consider: radical expressions with
one-term denominators and those with two-term denominators. For both, the
numerator and denominator will be multiplied by the same expression, which
is the same as multiplying by one.
EXAMPLE 1
Selecting a strategy to rationalize the denominator
3
Simplify 4兹 by rationalizing the denominator.
5
Solution
3
4V5
⫽
3
4V5
⫽
3 V5
4⫻5
⫽
3 V5
20
⫻
V5
V5
(Multiply both the numerator
and denominator by 兹5)
(Simplify)
When the denominator of a radical fraction is a two-term expression, you can
rationalize the denominator by multiplying by the conjugate.
An expression such as 兹a ⫹ 兹b has the conjugate Va ⫺ Vb.
Why are conjugates important? Recall that the linear terms are eliminated when
expanding a difference of squares. For example,
1a ⫺ b2 1a ⫹ b2 ⫽ a2 ⫹ ab ⫺ ab ⫺ b2
⫽ a2 ⫺ b2
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If a and b were radicals, squaring them would rationalize them.
QVm ⫹ VnR QVm ⫺ VnR, m, n rational
Consider this product:
⫽ Q兹mR 2 ⫺ 兹mn ⫹ 兹mn ⫺ Q兹nR 2
⫽m⫺n
Notice that the result is rational!
EXAMPLE 2
Creating an equivalent expression by rationalizing the denominator
Simplify
兹6
2
by rationalizing the denominator.
⫹ 兹3
Solution
2
V6 ⫹ V3
⫽
⫽
⫽
EXAMPLE 3
2
V6 ⫹ V3
⫻
2 Q 兹6 ⫺ 兹3R
V6 ⫺ V3
V6 ⫺ V3
(Multiply both the numerator and
denominator by 兹6 ⫺ 兹3)
(Simplify)
6⫺3
2 Q 兹6 ⫺ 兹3R
3
Selecting a strategy to rationalize the denominator
Simplify the radical expression
5
by rationalizing the denominator.
2兹6 ⫹ 3
Solution
5
2V6 ⫹ 3
⫽
⫽
⫽
⫽
⫽
5
2 V6 ⫹ 3
⫻
5 Q2 V6 ⫺ 3R
2 V6 ⫺ 3
2 V6 ⫺ 3
(The conjugate 2兹6 ⫹ 兹3 is 2兹6 ⫺ 兹3)
(Simplify)
4 V36 ⫺ 9
5 Q2 V6 ⫺ 3R
24 ⫺ 9
5 Q2 V6 ⫺ 3R
(Divide by the common factor of 5)
15
2V6 ⫺ 3
3
The numerator can also be rationalized in the same way as the denominator was
in the previous expressions.
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01_VCSB_Ch01_001-060.qxd
EXAMPLE 4
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Selecting a strategy to rationalize the numerator
Rationalize the numerator of the expression
兹7
⫺ 兹3
.
2
Solution
V7 ⫺ V3
V7 ⫺ V3
V7 ⫹ V3
⫽
⫻
2
2
V7 ⫹ V3
7⫺3
⫽
2 Q兹7 ⫹ 兹3R
4
⫽
2 Q兹7 ⫹ 兹3R
2
⫽
V7 ⫹ V3
(Multiply the numerator and
denominator by 兹7 ⫹ 兹3)
(Simplify)
(Divide by the common factor of 2)
IN SUMMARY
Key Ideas
• To rewrite a radical expression with a one-term radical in the denominator,
multiply the numerator and denominator by the one-term denominator.
兹a
兹a
兹b
⫽
⫻
兹b
兹b
兹b
兹ab
⫽
b
• When the denominator of a radical expression is a two-term expression,
rationalize the denominator by multiplying the numerator and denominator
by the conjugate, and then simplify.
1
兹a ⫺ 兹b
1
⫽
兹a ⫺ 兹b
兹a ⫹ 兹b
⫽
a⫺b
⫻
兹a ⫹ 兹b
兹a ⫹ 兹b
Need to Know
• When you simplify a radical expression such as 兹3 , multiply the numerator
5 兹2
and denominator by the radical only.
兹3
5兹2
⫻
兹2
兹2
⫽
⫽
兹6
5122
兹6
10
• 兹a ⫹ 兹b is the conjugate 兹a ⫺ 兹b, and vice versa.
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Exercise 1.1
PART A
1. Write the conjugate of each radical expression.
a. 2兹3 ⫺ 4
c. ⫺2 V3 ⫺ V2
e. V2 ⫺ V5
b. V3 ⫹ V2
d. 3 V3 ⫹ V2
f. ⫺V5 ⫹ 2V2
2. Rationalize the denominator of each expression. Write your answer in
simplest form.
a.
V3 ⫹ V5
c.
4 V3 ⫹ 3 V2
2 V3
V2
b.
2V3 ⫺ 3 V2
d.
3 V5 ⫺ V2
V2
K
2 V2
PART B
3. Rationalize each denominator.
a.
b.
3
V5 ⫺ V2
2V5
2V5 ⫹ 3 V2
c.
d.
V3 ⫺ V2
V3 ⫹ V2
2 V5 ⫺ 8
2 V5 ⫹ 3
e.
f.
2 V3 ⫺ V2
5 V2 ⫹ V3
3 V3 ⫺ 2V2
3 V3 ⫹ 2V2
4. Rationalize each numerator.
a.
C
V5 ⫺ 1
4
b.
2 ⫺ 3V2
2
c.
V5 ⫹ 2
2 V5 ⫺ 1
8兹2
.
⫺ 兹18
8兹2
b. Rationalize the denominator of
.
2兹5 ⫺ 3兹2
5. a. Rationalize the denominator of
兹20
c. Why are your answers in parts a and b the same? Explain.
6. Rationalize each denominator.
a.
b.
A
2V3 ⫺ V8
2V6
2V27 ⫺ V8
c.
d.
2 V2
V16 ⫺ V12
3 V2 ⫹ 2V3
V12 ⫺ V8
e.
f.
3 V5
4 V3 ⫺ 5V2
V18 ⫹ V12
V18 ⫺ V12
7. Rationalize the numerator of each of the following expressions:
a.
NEL
2 V2
兹a ⫺ 2
a⫺4
b.
Vx ⫹ 4 ⫺ 2
x
c.
兹x ⫹ h ⫺ 兹x
h
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