Surds and Algebraic Fractions

Step 2: Apply the multiplication rule and simplify.

• $\sqrt{54} = \sqrt{9} \times \sqrt{6} = 3\sqrt{6}$
• $\sqrt{24} = \sqrt{4} \times \sqrt{6} = 2\sqrt{6}$

Step 3: Subtract the like surds.

• $3\sqrt{6} - 2\sqrt{6} = \sqrt{6}$

Rationalising the Denominator

Why do mathematicians hate leaving square roots at the bottom of a fraction? It makes division extremely difficult, so we use a process called rationalising to convert an irrational denominator into a rational number.

For simple fractions like $\frac{1}{\sqrt{a}}$, multiply both the numerator and denominator by $\sqrt{a}$. For complex fractions, you must multiply by the conjugate. The conjugate is the exact same denominator expression but with the sign reversed (e.g., the conjugate of $\sqrt{a} + b$ is $\sqrt{a} - b$). This applies the Difference of Two Squares (DOTS) principle, which cancels out the surd:

$$(x+y)(x-y) = x^2 - y^2$$

Rationalise and simplify $\frac{4}{\sqrt{6}-2}$

Step 1: Multiply the numerator and denominator by the conjugate of the denominator.

• $\frac{4}{\sqrt{6}-2} \times \frac{\sqrt{6}+2}{\sqrt{6}+2}$

Step 2: Expand the denominator using the DOTS principle.

• $(\sqrt{6} - 2)(\sqrt{6} + 2) = (\sqrt{6})^2 - 2^2 = 6 - 4 = 2$

Step 3: Substitute the new denominator and simplify the fraction.

• $\frac{4(\sqrt{6} + 2)}{2}$
• $2(\sqrt{6} + 2) = 2\sqrt{6} + 4$

Multiplying and Dividing Algebraic Fractions

Multiplying algebraic fractions is often easier than adding them, because you do not need a common denominator. The fundamental rule is to always factorise numerators and denominators fully before attempting to multiply or divide.

For multiplication, use the rule $\frac{A}{B} \times \frac{C}{D} = \frac{AC}{BD}$. Common factors can then be cancelled between any numerator and any denominator. For division, multiply the first fraction by the reciprocal of the second fraction (often called "Keep, Flip, Change"). Continue canceling until the fraction is in its simplest form.

Simplify $\frac{x^2 - 25}{2x + 10} \div \frac{x - 5}{4}$

Step 1: Rewrite the division as multiplication using the reciprocal.

• $\frac{x^2 - 25}{2x + 10} \times \frac{4}{x - 5}$

Step 2: Factorise all numerators and denominators fully.

• $x^2 - 25$ is DOTS: $(x - 5)(x + 5)$
• $2x + 10$ is a single bracket: $2(x + 5)$
• Expression becomes: $\frac{(x - 5)(x + 5)}{2(x + 5)} \times \frac{4}{x - 5}$

Step 3: Cancel common factors and calculate the final answer.

• Cancel the $(x - 5)$ and $(x + 5)$ terms from the top and bottom.
• $\frac{4}{2} = 2$

Adding and Subtracting Algebraic Fractions

Understanding how to combine fractions is essential for solving complex algebraic equations later on. Unlike multiplication, you must find a common denominator before you can combine the numerators.

Find a common denominator by finding the Lowest Common Multiple (LCM) or multiplying the denominators together. If working on a Higher Tier paper, factorise quadratic denominators first to avoid creating overly complex cubic expressions. When adjusting numerators, always use brackets to ensure you multiply all terms correctly.

Express $\frac{4}{x^2-9} + \frac{1}{x+3}$ as a single fraction in its simplest form.

Step 1: Factorise the denominators to identify the lowest common denominator.

• $x^2 - 9 = (x + 3)(x - 3)$
• The lowest common denominator is $(x + 3)(x - 3)$.

Step 2: Adjust the numerators to match the common denominator.

• The first fraction already has the common denominator: $\frac{4}{(x+3)(x-3)}$
• The second fraction needs to be multiplied by $(x - 3)$ on top and bottom: $\frac{1(x - 3)}{(x + 3)(x - 3)}$

Step 3: Combine the numerators and simplify.

• $\frac{4 + 1(x - 3)}{(x + 3)(x - 3)}$
• $\frac{4 + x - 3}{(x + 3)(x - 3)} = \frac{x + 1}{(x + 3)(x - 3)}$