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MATH 31 LESSONS
PreCalculus
3. Simplifying Rational Expressions
Rationalization
A. Rational Expressions
To simplify a rational expression:
Factor everything in the numerator and denominator
Cancel any shared factor between the top and the bottom
Leave the answer in factored form
e.g. Simplifyxxx
xx
54
20923
2
xxx
xx
54
20923
2
54
452
xxx
xxFactor the top and the bottom
xxx
xx
54
20923
2
54
452
xxx
xx
15
45
xxx
xxBe certain they are fully factored
xxx
xx
54
20923
2
54
452
xxx
xx
1
4
xx
xCancel any shared factor between the top and bottom
15
45
xxx
xx
Multiplying Rational Expressions
Let’s start by reviewing how to multiply fractions.
Try multiplying4
3
7
5
4
3
7
5
47
35
When you multiply fractions, you multiply:
top top
bottom bottom
4
3
7
5
47
35
28
15
Now we will try multiplying rational expressions.
Multiply and simplify
4
2
4
162
23
4
2
x
xx
xx
x
4
2
4
162
23
34
2
x
xx
xx
x
22
2
4
44 2
3
xx
xx
xx
xx
Fully factor all expressions
4
2
4
162
23
34
2
x
xx
xx
x
22
2
4
44 2
3
xx
xx
xx
xx
422
4423
2
xxxx
xxxx
Multiply: top top, bottom bottom
Note: This is an optional step.You don’t have to do it.
4
2
4
162
23
34
2
x
xx
xx
x
22
2
4
44 2
3
xx
xx
xx
xx
422
4423
2
xxxx
xxxx
2
4
xx
x
Cancel any shared factors between the top and bottom
Dividing Rational Expressions
Let’s start by reviewing how to divide fractions.
Try dividing5
11
3
7
5
11
3
7
11
5
3
7
When you divide a fraction, you “invert and multiply”
5
11
3
7
11
5
3
7
33
35 When you multiply fractions, it is:
top topbottom bottom
Now we will try dividing rational expressions.
Divide and simplify
aa
aa
aa
aa
63
44
78
1452
2
2
2
aa
aa
aa
aa
63
44
78
1452
2
2
2
44
63
78
1452
2
2
2
aa
aa
aa
aa
When you divide a fraction, you “invert and multiply”
aa
aa
aa
aa
63
44
78
1452
2
2
2
44
63
78
1452
2
2
2
aa
aa
aa
aa
22
23
17
27
aa
aa
aa
aa
Fully factor all expressions
aa
aa
aa
aa
63
44
78
1452
2
2
2
44
63
78
1452
2
2
2
aa
aa
aa
aa
22
23
17
27
aa
aa
aa
aa
1
3
a
aThere is no need to multiply them.
Simply cancel any shared factors on the top and bottom.
Adding and Subtracting Rational Expressions
Let’s start by reviewing how to add and subtract fractions.
Try simplifying4
5
2
1
3
7
4
5
2
1
3
7
3
3
4
5
6
6
2
1
4
4
3
7
LCD: 12
Multiply the top and bottom of each fraction by the number which will make the LCD
4
5
2
1
3
7
3
3
4
5
6
6
2
1
4
4
3
7
12
15
12
6
12
28
4
5
2
1
3
7
3
3
4
5
6
6
2
1
4
4
3
7
12
15
12
6
12
28
12
15628 When they have the same
denominator, you add / subtract the numerators (while keeping the denominator the same)
4
5
2
1
3
7
3
3
4
5
6
6
2
1
4
4
3
7
12
15
12
6
12
28
12
15628
12
19
Adding and Subtracting Rational Expressions
For rational expressions, use the following procedure:
Find the LCD
- you may need to factor the denominators first
- the LCD of the variables is the largest power
For each fraction, multiply the top and bottom with
the “missing factor” to create the LCD
Add or subtract the numerator, while leaving the
denominator the same
e.g. 1 Simplify xyx 8
5
6
12
xyx 8
5
6
12
x
x
xyy
y
x 3
3
8
5
4
4
6
12
LCD: 24x2y
Multiply the top and bottom of each fraction with what’s needed to make the LCD
xyx 8
5
6
12
x
x
xyy
y
x 3
3
8
5
4
4
6
12
yx
x
yx
y22 24
15
24
4
xyx 8
5
6
12
x
x
xyy
y
x 3
3
8
5
4
4
6
12
yx
x
yx
y22 24
15
24
4
yx
xy224
154
When the denominator is the same, you can add the numerators (while leaving the denominator the same)
e.g. 2 Simplify 32
2
3
522
xx
x
xx
x
32
2
3
522
xx
x
xx
x
13
2
3
5
xx
x
xx
x
Factor the denominators
32
2
3
522
xx
x
xx
x
13
2
3
5
xx
x
xx
x
xx
xx
x
x
x
xx
x
13
2
1
1
3
5
LCD: x (x - 3) (x + 1
Multiply the top and bottom of each fraction with the “missing factors”
32
2
3
522
xx
x
xx
x
13
2
3
5
xx
x
xx
x
xx
xx
x
x
x
xx
x
13
2
1
1
3
5
13
215
xxx
xxxx Since the denominators are the same, you can subtract the numerators (and keep the denominator the same)
13
215
xxx
xxxx
13
256 22
xxx
xxxx
Simplify the numerator
13
215
xxx
xxxx
13
256 22
xxx
xxxx
13
58
xxx
x
Keep the denominator factored
Ex. 1 Simplify 12
20
16
1242
2
2
2
xx
x
x
xx
Try this example on your own first.Then, check out the solution.
12
20
16
1242
2
2
2
xx
x
x
xx
2
2
2
2
20
12
16
124
x
xx
x
xx
When you divide a fraction, invert and multiply.
Also, put the polynomials in the same order (x’s first, constant last)
12
20
16
1242
2
2
2
xx
x
x
xx
2
2
2
2
20
12
16
124
x
xx
x
xx
22 20
34
16
34
x
xx
x
xx
Factor all expressions
12
20
16
1242
2
2
2
xx
x
x
xx
2
2
2
2
20
12
16
124
x
xx
x
xx
22 20
34
16
34
x
xx
x
xx
220
34
44
34
x
xx
xx
xx
220
34
44
34
x
xx
xx
xx
Cancel any shared factors between the numerator and the denominator
420
342
2
xx
xx
220
34
44
34
x
xx
xx
xx
420
342
2
xx
xx
45
3 2
xx
x
12
20
16
1242
2
2
2
xx
x
x
xx
2
2
2
2
20
12
16
124
x
xx
x
xx
22 20
34
16
34
x
xx
x
xx
220
34
44
34
x
xx
xx
xx
45
3 2
xx
x
Cancel any shared factors between the numerator and the denominator
Ex. 2 Simplify xxxx 10
2
5015
122
Try this example on your own first.Then, check out the solution.
xxxx 10
2
5015
122
10
2
510
1
xxxx
Factor the denominators
xxxx 10
2
5015
122
10
2
510
1
xxxx
5
5
10
2
510
1
x
x
xxx
x
xx
LCD: x (x - 10) (x - 5)
Multiply the top and bottom of the fractions with the “missing factors”
xxxx 10
2
5015
122
10
2
510
1
xxxx
5
5
10
2
510
1
x
x
xxx
x
xx
510
52
xxx
xxWhen the denominators are the same, simply subtract the numerators (leaving the denominator the same)
510
52
xxx
xx
510
102
xxx
xx
Simplify the numerator, but leave the denominator factored
510
52
xxx
xx
510
102
xxx
xx
510
10
xxx
x
Don’t stop yet.
What else can be done?
510
52
xxx
xx
510
102
xxx
xx
510
10
xxx
x
510
10
xxx
x
510
52
xxx
xx
510
102
xxx
xx
510
10
xxx
x
510
10
xxx
x 5
1
xx
Ex. 3 Simplify
14
4
13
32
xx
xx
x
Try this example on your own first.Then, check out the solution.
14
4
13
32
xx
xx
x
14
11
14
13
11
13 2
xxxx
xx
xxx
Bring the fractions together (using LCD) on top and bottom
14
4
13
32
xx
xx
x
14
11
14
13
11
13 2
xxxx
xx
xxx
1
4141
313 2
xxxx
xxx
1
4141
313 2
xxxx
xxx
414
1
1
313 2
xx
x
x
xxx
When you divide by a fraction, invert and multiply
1
4141
313 2
xxxx
xxx
414
1
1
313 2
xx
x
x
xxx
445
3322
22
xx
xxx
1
4141
313 2
xxxx
xxx
414
1
1
313 2
xx
x
x
xxx
445
3322
22
xx
xxx
xx
xx
5
6222
2
xx
xx
5
6222
2
5
32 2
xx
xx
After factoring, there are no shared factors. It cannot be simplified further.
B. Rationalizing Radical Expressions
B1. Radical Arithmetic
If a > 0,
?2 a
If a > 0,
aa 2
Squaring and square-rooting are “opposites” (inverses), and so, they cancel each other out
e.g. Express as a simplified mixed radical
725
725
2365 Factor out the largest perfect square in the radicand
725
2365
Separate the radicals2365
725
2365
2365
265
725
2365
2365
265
230
Radical Arithmetic
If c > 0,
? cbca
If c > 0,
cbacbca
If they are like radicals, you can add (or subtract) the coefficients. The radical stays the same.
e.g. Simplify 3611538117
3611538117
3638115117
3611538117
3638115117
321112
Radical Arithmetic
If a, b > 0,
? ba
If a, b > 0,
abba
When you multiply radicals, you simply multiply the radicands (the numbers inside the radical)
If a, b > 0,
Similarly,
abba
b
a
b
a bdacdcba
e.g. Simplify 21
423765
21
423765
21
421835
21
423765
21
421835
22335
21
423765
21
421835
22335
22105
21
423765
21
421835
22335
22105 2106
Ex. 4 Simplify 26534
Try this example on your own first.Then, check out the solution.
26534
65346534
It is recommended that you write it out twice.
Remember: (a + b)2 ≠ a2 + b2
26534
65346534
22653465346534
Use FOIL to expand.
26534
65346534
22653465346534
362518201820916
26534
65346534
22653465346534
362518201820916
1840625316 Bring together like radicals
1840625316
234015048
1840625316
234015048
2120198
B2. Rationalizing Denominators
When you rationalize a denominator, you are removing all radicals from the denominator.
We will consider two cases:
1. Monomial denominators
2. Binomial denominators
Case 1: Monomial denominators
Use the following procedure:
Simplify the radicals as much as possible
- no perfect squares in the radicand
Multiply the top and bottom by the radical in
the denominator
e.g. Rationalize the denominator for
20
35
20
35
52
35
Simplify all radicals first
20
35
52
35
5
5
52
35
Multiply the top and bottom by the radical in the denominator.
No need to multiply by the coefficient, since it is already rational
20
35
52
35
5
5
52
35
252
155
If rationalization is done properly, the radical becomes squared
20
35
52
35
5
5
52
35
252
155
52
155
Now, there are no radicals in
the denominator. It has been rationalized.
20
35
52
35
5
5
52
35
252
155
52
155
2
15
Case 2: Binomial denominators
Use the following procedure:
Multiply the top and bottom by the “conjugate” of the
denominator
“a + b” and “a b” are called conjugates
This should result in a difference of squares (a2 - b2),
which removes the radicals
e.g. 1 Rationalize the denominator for
25
6
25
6
25
25
25
6
Multiply the top and bottom by the “conjugate” of the denominator.
If the denominator is of the form “a - b”, then the conjugate is “a + b”
25
6
25
25
25
6
22
25
256
If done properly, each of the radicals in the denominator will be squared.
(a + b) (a - b) = a2 - b2
25
6
25
25
25
6
22
25
256
25
1230
Now, there are no radicals in the denominator. It has been rationalized.
25
6
25
25
25
6
22
25
256
25
1230
3
3230
e.g. 2 Rationalize the denominator for
xx 4
122
xx 4
122
xx
xx
xx
4
4
4
122
2
2
Multiply the top and bottom by the conjugate of the denominator.
If the denominator is of the form “a + b”, then the conjugate is “a - b”
xx 4
122
xx
xx
xx
4
4
4
122
2
2
2
22
2
4
412
xx
xx
This results in the radical being squared.
(a + b) (a - b) = a2 - b2
xx 4
122
xx
xx
xx
4
4
4
122
2
2
2
22
2
4
412
xx
xx
22
2
4
412
xx
xx
There are no radicals in the denominator. It has been rationalized.
22
2
4
412
xx
xx
4
412 2
xx
Simplify the denominator, but leave the numerator factored. There may be an opportunity to simplify.
22
2
4
412
xx
xx
4
412 2
xx
xx 43 2
Ex. 5 Rationalize the denominator for
153
12
Try this example on your own first.Then, check out the solution.
153
12
153
153
153
12
Multiply the top and bottom by the conjugate.
If the denominator is of the form “a + b”, then the conjugate is “a - b”
153
12
153
153
153
12
22
153
15312
The radical in the denominator should become squared.
(a + b) (a - b) = a2 - b2
153
12
153
153
153
12
22
153
15312
159
15312
There are no radicals in the denominator. It has been rationalized.
159
15312
44
15312
Leave the top factored. There may be an opportunity to simplify.
159
15312
44
15312
11
1533
Ex. 5 Rationalize the numerator for
3
32 2
x
xxx
Try this example on your own first.Then, check out the solution.
3
32 2
x
xxx
xxx
xxx
x
xxx
32
32
3
322
22
Multiply the top and bottom by the conjugate of the numerator.
If the numerator is of the form “a - b”, then the conjugate is “a + b”
3
32 2
x
xxx
xxx
xxx
x
xxx
32
32
3
322
22
xxxx
xxx
323
322
22
2
If done properly, the radical in the numerator will be squared.
(a - b) (a + b) = a2 - b2
3
32 2
x
xxx
xxx
xxx
x
xxx
32
32
3
322
22
xxxx
xxx
323
322
22
2
xxxx
xxx
323
322
22 Do not simplify the denominator. There may be an opportunity to simplify.
xxxx
xxx
323
322
22
xxxx
xx
323
32
2
xxxx
xxx
323
322
22
xxxx
xx
323
32
2
xxxx
xx
323
32
xxxx
xxx
323
322
22
xxxx
xx
323
32
2
xxx
x
32 2
xxxx
xx
323
32
Ex. 6 Rationalize the numerator for
h
xhx 12122
Try this example on your own first.Then, check out the solution.
h
xhx 12122
12122
1212212122
xhx
xhx
h
xhx
Multiply the top and bottom by the conjugate of the numerator.
If the numerator is of the form “a - b”, then the conjugate is “a + b”
h
xhx 12122
12122
1212212122
xhx
xhx
h
xhx
12122
1212222
xhxh
xhx
If done properly, the radicals will become squared.
(a - b) (a + b) = a2 - b2
h
xhx 12122
12122
1212212122
xhx
xhx
h
xhx
12122
1212222
xhxh
xhx
12122
12122
xhxh
xhxThe numerator no longer has a radical. It has been rationalized.
12122
12122
xhxh
xhx
12122
12122
xhxh
xhx
Do not simplify the denominator. There may be an opportunity to cancel.
12122
12122
xhxh
xhx
12122
12122
xhxh
xhx
12122
2
xhxh
h
12122
12122
xhxh
xhx
12122
12122
xhxh
xhx
12122
2
xhxh
h
12122
2
xhx
