HFCC Math Lab
Intermediate Algebra - 17
DIVIDING RADICALS AND RATIONALIZING THE DENOMINATOR
Dividing Radicals: To divide radical expression we use
Step 1: Simplify each radical
Step 2: Apply the Quotient rule for Radicals
Rule1:
& Rule2:
Step 3: After applying the rule simplify the expression if possible.
Ex 1: Simplify
Applying Rule 1:
Simplify the expression inside the radicals:
Ex 2: Simplify
Applying Rule 2:
Simplify the expression inside the radical:
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Ex 3: Simplify
Simplifying:
Applying Rule 1:
Simplify the expression inside the radicals
Ex 4: Simplify
Simplifying the expression inside the radicals
Applying Rule 2:
Simplify the expression inside the radical
Exercises: Divide and simplify using the following radical expressions
1.
2.
3.
4.
5.
6.
7.
8.
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Rationalizing the Denominator: In order for a radical expression to be in the simplest radical form,
there can be no fractions under the radical sign and no radicals in the denominator of the fraction. The
process of removing the radicals from the denominator is called rationalizing the denominator.
A. Rationalizing the denominator when the denominator has only one term with index n
Step 1: Simplify any radicals in the numerator and the denominator.
Step 2: Determine the factor needed to make the denominator radicand a perfect nth power
Step 3: Multiply the numerator and denominator with the factor determined in step 2
Step 4: Simplify the resulting expression if possible
Ex 5: Rationalize the denominator
Step 1: Simplify the radical
Step 2: The factor required to make the denominator radicand a perfect 2nd power is
Step 3: Multiplying the numerator and denominator by
:
Step 4: Simplify the resulting expression:
Ex 6: Rationalize the denominator
Step 1: Simplify the radical
Step 2: The factor required to make the denominator radicand a perfect 3rd power is
Step 3: Multiplying the numerator and denominator by
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:
Step 4: Simplify the resulting expression:
Ex 7: Rationalize the denominator
Step 1: Simplify the radical
Step 2: The factor required to make the denominator radicand a perfect 3rd power is
Step 3: Multiplying the numerator and denominator by
:
Step 4: Simplify the resulting expression:
Ex 8: Rationalize the denominator
Step 1: Simplify the radical
Step 2: The factor required to make the denominator radicand a perfect 3rd power is
Step 3: Multiplying the numerator and denominator by
Step 4: Simplify the resulting expression:
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:
=
B. Rationalizing the denominator with two terms, one or both of which involve square root.
Recall that the binomials
and
are called the conjugates. And the difference of square
formula the product of the conjugates will result in
.
Step 1: Multiply the numerator and denominator by the conjugate of the denominator
Step 2: Simplify the resulting expression if possible
Ex 9: Rationalize the denominator
Step1: Multiplying the numerator and denominator with the conjugate
of the
denominator:
Step 2: Simplify:
Ex 10: Rationalize the denominator
Step1: Multiplying the numerator and denominator with the conjugate
of the
denominator:
Step 2: Simplify:
Ex 11: Rationalize the denominator
Step1: Multiplying the numerator and denominator with the conjugate
denominator:
Step 2: Simplify:
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of the
Exercises: Rationalize the denominator in each of the following. Assume that the variable are
positive and the denominator is equal to zero
9.
10.
12.
11.
13.
14.
15.
16.
18.
17.
19.
20.
21.
22.
23.
24.
Solutions to the Exercise problems:
Exercises: Divide and simplify using the following radical expressions
1.
=
=
2.
=
=
=
=
=
4.
3.
=
=
=
=
=
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5.
=
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6.
=
=
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7.
8.
=
=
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9.
10.
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12.
11.
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13.
14.
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=
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7
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15.
16.
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18.
17.
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19.
20.
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21.
22.
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23.
24.
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=
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You can get additional instruction and practice by going to the following website
http://www.purplemath.com/modules/radicals5.htm This website provides good
review and practice problems for quotient rule and rationalizing the denominator
http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut40
_addrad.htm This website provides good review and practice for rationalizing the
denominator
http://www.helpalgebra.com/articles/rationalizedenominator.htm This website
provides good review and practice for rationalizing the denominator
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