Manipulating Algebraic Fractions
# Manipulating Algebraic Fractions

GCSE(H),

An algebraic fraction is a fraction with an unknown. Simplifying an algebraic fraction involves determining common factors and dividing the numerator and the denominator by those factors.

Given `frac(x^2 + 8x + 16)(x^2 - 16)`.

Factorise the numerator and the denominator to get `frac((x + 4)(x + 4))((x + 4)(x - 4))`.

The (*x* + 4) term is common to both the numerator and the denominator. Simplify to `frac(x + 4)(x - 4)`.

When calculating with algebraic fractions, the same rules apply as they do for arithmetic fractions: when adding and subtracting, the denominators must be the same; multiplying involves multiplying the numerators together and multiplying the denominators together; and dividing involves inverting the second fraction, then multiplying the two fractions.

For example, simplify `1/(a + 4) + 3/(a - 4)`

= `frac((a - 4))((a + 4)(a - 4))` + `frac(3(a + 4))((a - 4)(a + 4))`

= `frac(a - 4 + 3a + 12)((a + 4)(a - 4))`

= `frac(4(a + 2))((a + 4)(a - 4))`

## Examples

1. Simplify `frac((a^2 + 4a + 4))((a^2 - 4))`.

Answer: `frac((a + 2))((a - 2))`

`frac((a^2 + 4a + 4))((a^2 - 4))` = `frac((a + 2)(a + 2))((a + 2)(a - 2))` = `frac((a + 2))((a - 2))`

2. Calculate `frac((x - 3))((x + 3))` x `frac((4x + 12))((2x - 6))`.

Answer: 2

`frac((x - 3))((x + 3))` x `frac((4x + 12))((2x - 6))` = `frac(4(x - 3)(x + 3))(2(x + 3)(x - 3))` = `frac(4)(2)` = 2

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