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Working with Surds

You can write down the exact length of a diagonal using a square root, but as a decimal, those numbers never end. A surd provides an exact value by keeping the square root of a non-square number, resulting in an irrational number. Using exact values is crucial in Edexcel exams to prevent accuracy loss from rounding.

To simplify surds, use the core multiplication rule: $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$ . You must identify a factor of the that is the largest possible square number (e.g., ). You can only add or subtract "like" surds, which are surds that have the identical radicand.

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 . The conjugate is the exact same denominator expression but with the sign reversed (e.g., the conjugate of is ). 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.

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

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 of the second fraction (often called "Keep, Flip, Change"). Continue canceling until the fraction is in its .

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:

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:

Step 3: Combine the numerators and simplify.

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

Students often assume that $\sqrt{a} + \sqrt{b} = \sqrt{a+b}$ , but this is strictly incorrect; only 'like' surds with identical radicands can be combined.

In 'Show That' questions on Edexcel calculator papers, examiners require you to explicitly write out all intermediate simplification and rationalisation steps to earn full marks.

When subtracting algebraic fractions, remember to distribute the negative sign across every term in the second numerator (e.g., $-\frac{x-3}{w}$ becomes $\frac{-x+3}{w}$ ).

Expect terms to cancel! If you cannot find any common factors to cancel in an algebraic fraction multiplication question, go back and double-check your quadratic factorisation.

Examiners prefer that you leave the final denominator of algebraic fractions in its factorised form (e.g., $(x+3)(x-3)$ ) rather than expanding it back out.

The square root of a number that is not a square number, leaving an exact value that is an irrational number.

A decimal that never ends and never repeats, which cannot be expressed as a fraction of integers.

The number or expression located underneath the square root symbol.

The mathematical process of removing a surd from the denominator to create an equivalent fraction with an integer denominator.

A number that can be expressed exactly as a fraction of two integers.

An expression with the sign between its terms reversed, used to rationalise denominators with multiple terms (e.g., changing a + b to a - b).

The result of inverting a fraction, meaning the numerator and denominator swap places.

A fraction that has been fully reduced, leaving no common factors between the numerator and the denominator.