Simplifying Algebraic Fractions (Cambridge International IGCSE Maths)

Simplifying Algebraic Fractions

What is an algebraic fraction?

• A fraction with a numerator or denominator (or both) that contains an algebraic term is an algebraic fraction

• For example

  • $\frac{x}{4}$ or $\frac{48}{y^{2}}$ or $\frac{2p^{3}}{4p^{2}+1}$

• You need to be able to simplify basic algebraic fractions

  • Like equation$\frac{x^{5}}{x^{2}}$ or $\frac{3}{6p}$

How do I simplify algebraic fractions?

• To simplify any powers(indices) it helps to write out the multiplications in full

  • $\frac{n^{5}}{n^{2}}=\frac{n\times n\times n\times n\times n}{n\times n}$

• Terms can then be "cancelled"

  • equation``n ^ 5 / n ^2 = (n x n x n x / n x / n)/(/ n x / n) = n x n x n = n ^3
  • $\frac{m^{2}}{m^{3}} = \frac{m \times m}{m \times m \times m} = \frac{ \cancel{m} \times \cancel{m} }{m \times \cancel{m} \times \cancel{m} } = \frac{1}{m}$

• Any numerical fractions can be simplified as usual

  • $\frac{4}{12}=\frac{2}{6}=\frac{1}{3}$

• You also need to be familiar with alternative ways of writing fractions

  • $\frac{3x}{4}$ is the same as $3\times\frac{x}{4}$ or $\frac{3}{4}x$ or $0.75x$
  • $\frac{5}{6y}$ is the same as $5\times\frac{1}{6y}$ or $\frac{5}{6}\times\frac{1}{y}$

Worked Example Simplify equationthe following expressions.

(a) $\frac{4p^{3}}{12p^{6}}$

Separate the numerical part from the algebraic part

$\frac{4}{12}\times\frac{p^{3}}{p^{6}}$

Simplify the numerical fraction

$\frac{1}{3}\times\frac{p^{3}}{p^{6}}$

Write out the algebraic terms in full

$\frac{1}{3}\times\frac{p\times p\times p}{p\times p\times p\times p\times p\times p}$

Cancel any terms that appear in both the numerator and denominator

equation$1 \frac{1}{3} \times \frac{ \cancel{p} \times \cancel{p} \times \cancel{p} }{p \times p \times p \times \cancel{p} \times \cancel{p} \times \cancel{p} }$

$\frac{1}{3}\times\frac{1}{p\times p\times p}$

Simplify

$\frac{1}{3}\times\frac{1}{p^{3}}$

Combine into one term

$\frac{1}{3p^{3}}$

(b) $\frac{32q^{5}}{8q^{2}}$

Separate the numerical part from the algebraic part

$\frac{32}{8}\times\frac{q^{5}}{q^{2}}$

Simplify the numerical fraction

equation$4\times\frac{q^{5}}{q^{2}}$

Write out the algebraic terms in full

$4\times\frac{q\times q\times q\times q\times q}{q\times q}$

Cancel any terms that appear in both the numerator and denominator

$4 \times \frac{q \times q \times q \times \cancel{q} \times \cancel{q} }{ \cancel{q} \times \cancel{q} }$

$4\times q\times q\times q$

Simplify

$4q^{3}$