Building Up the Denominator
Only rational expressions with identical denominators
can be added or subtracted. Fractions without identical denominators can be
converted to equivalent fractions with a common denominator by reversing the
procedure for reducing fractions to its lowest terms. This procedure is called
building up the denominator.
Consider converting the fraction
into an equivalent fraction with a denominator
of 51. Any fraction that is equivalent to
can be obtained by multiplying the numerator
and denominator of
by the same nonzero number. Because 51
= 3 Â· 17, we multiply the numerator and denominator of
by 17 to get an equivalent fraction
with a denominator of 51:
Example 1
Building up the denominator
Convert each rational expression into an equivalent rational expression that has the
indicated denominator.
Solution
a) Factor 42 as 42 = 2 Â· 3 Â· 7, then multiply the numerator and denominator of
by the missing factors, 2 and 3:
b) Because 9a^{3}b^{4} = 3ab^{3} Â· 3a^{2}b, we multiply the numerator and denominator by
3ab^{3}:
When building up a denominator to match a more complicated denominator, we
factor both denominators completely to see which factors are missing from the simpler
denominator. Then we multiply the numerator and denominator of the simpler
expression by the missing factors.
Helpful hint
Notice that reducing and
building up are exactly the
opposite of each other. In reducing
you remove a factor
that is common to the numerator
and denominator, and in
buildingup you put a common
factor into the numerator
and denominator.
Example 2
Building up the denominator
Convert each rational expression into an equivalent rational expression that has the
indicated denominator.
Solution
a) Factor both 2a  2b and 6b  6a to see which factor is missing in 2a  2b.
Note that we factor out 6 from 6b  6a to get the factor a  b:
2a  2b 
= 2(a  b) 
6b  6a 
= 6(a  b) = 3 Â· 2(a  b) 
Now multiply the numerator and denominator by the missing factor, 3:
b) Because x^{2} + 7x + 12 = (x + 3)(x + 4), multiply the numerator and denominator
by x + 4:
Helpful hint
Multiplying the numerator
and denominator of a rational
expression by 1 changes the
appearance of the expression:
