Same rules as arithmetic operations on fractions.
Addition and Subtraction
12a2bx−18ab2y
- Find LCD
- For each denominator find the required factors to make them equal to the LCD
- Multiply the numerators by these factors
- Add or subtract the resulting numerators on top of the LCD
Step 1
36a2b2
Step 2
- Term: 3b
- Term: 2a
Step 3
- Term: x×3×b
- Term: y×2×a
Step 4
This yields the result of the above expression: 36a2b23bx−2ya
Alternatively multiply the numerators by the LCD, leaving the denominators alone, simplify the fractions and put the resulting terms on top of the LCD.
Multiplication and Division
In case of multiplication simply multiply both the terms of both fractions. In case of division multiply the reciprocal of one term with the other.
14x2y26ax4×3a42y3
=42a4x2y212ax4y3
simplify (variable with negative exponent is moved to the other side of the fraction with the sign reversed)
42a4x2y212ax4y3=7a32x2y
Taking the reciprocal of both sides
If two fractions are equal, their reciprocates are also equal. This can be used in solving equations involving fractions. Take for example
V=31πr2h and solve for h
Dividing by h yields hV=31πr2 which is equal to hV=3πr2
Now V needs to be moved to the denominator so that it can be moved to the other side by multiplication. This can be achieved by taking the reciprocal of both sides of the equation:
hV=3πr2 is the same as Vh=πr23
From that equation multiply by V to solve for h:
h=πr23V
Complex Problems
Distributing Results of Operations
12(2a−b−3a+b)
- Subtract fractions within brackets
- Multiply the result with the coefficient
- Simplify (if the denominator is a common factor with both numerators)
Step 1
63a−3b−62a+2b=63a−3b+6−2a−2b
=6a−5b
Step 2
112×6a−5b=612a−60b
Step 3
12a−10b=2a−10b
Distribution of exponents
(−x)3=(1−x)3=3−3x3
however
x3=1x3
Factoring
Expressions may be factored to isolate subjects:
P+21bP=Q(b+21)
P(1+21b)=Q(b+21)
Moving factors outside of fractions
If the numerator is a single term expression, any of its factors may be pushed out to become factors of the whole fraction:
cdab=b(cda)
P(1+21b)=Q(b+21)
P=1+21bQ(b+21)
P=1+21bb+21Q
Cancelling
Only factors may be cancelled, never whole terms
x+3x+6=12
A factor may only cancel out with one other factor
b(a+2b)(a+3b)a(a+3b)(a+3b)=b(a+2b)a(a+3b)
Complex Fractions
cba=ba×c1=bca