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Mixed Expressions

A mixed expression, commonly called a complex fraction , is a fraction where the numerator, the denominator or both the numerator and denominator contain a fraction.

We can use the rules of algebra to manipulate these expressions into rational ones.

Example 1:

Simplify:

( 3 x x + 1 ) ( 2 + x 4 x 2 )

First, rewrite the denominator as a single rational expression.

( 3 x x + 1 ) ( 2 + x 4 x 2 ) = ( 3 x x + 1 ) ( ( 8 x 4 ) + x 4 x 2 ) = ( 3 x x + 1 ) ( 9 x 4 4 x 2 )

Since the fraction line means divide, rewrite the problem as multiplication by the reciprocal .

= 3 x x + 1 4 x 2 9 x 4 = 12 x 2 6 x 9 x 2 + 5 x 4

This is a rational expression.

A second method we can use is to multiply the numerator and denominator of the original improper fraction by the LCD of the two fractions (which will be another name for 1 ).

Example 2:

Simplify:

( 3 x x + 1 ) ( 5 2 x )

Multiply both numerator and denominator by 2 x ( x + 1 ) .

= ( 3 x x + 1 2 x ( x + 1 ) ) ( 5 2 x 2 x ( x + 1 ) ) = 3 x 2 x 5 ( x + 1 ) = 6 x 2 5 x + 5