# Adding and subtracting rational expressions - Higher

Adding and subtracting algebraic fractions is a similar process to adding and subtracting normal fractions.

Fractions can only be added or subtracted when there is a and algebraic fractions are the same in this method.

### Example

Write as a single fraction.

In this example, the of the two fractions are the same, so the can simply be added.

This gives .

This fraction cannot be simplified so is the final answer.

Question

Write as a single fraction.

The denominators of each fraction are different, and , so a common denominator must be created. This is found by working out the lowest common multiple of and which is .

Remember, if it is difficult to work out the lowest common multiple of two expressions, a common denominator can always be found by simply multiplying the denominators together. Remember this may mean the fractions need simplifying at the end.

To create a common denominator of , the first fraction's numerator and denominator must be multiplied by 7 and the second fraction's numerator and denominator must be multiplied by 3:

Now the denominators are the same, the numerators can be subtracted:

This fraction cannot be simplified any further, so is the final answer.

Question

Simplify .

These fractions do not have a common denominator. There are also no common factors between the denominators, so the only way to create a common denominator is to multiply the two expressions together.

can be written as .

(multiply numerator and denominator by ) (multiply numerator and denominator by )

Expanding brackets means everything inside the bracket has to be multiplied by the term outside the bracket.

This gives .

Now the denominators are the same, the numerators can be simplified:

Collect the like terms: . . .

This fraction cannot be simplified, so this is the final answer.