Welcome to the World of Surds! (Extended Content E1.17)

Hello! This chapter is where we learn how to handle those tricky square roots that don't simplify neatly. These numbers are called Surds (also known as radical expressions). They are part of the 'Number' section and are crucial for ensuring your answers are exact, rather than rounded decimals.

Don't worry if these look tricky at first—surds follow a very strict set of rules, and once you master these rules, they are just like handling algebraic terms!

What Exactly Is a Surd?

Definition and Context

A surd is the mathematical name given to an irrational root of a rational number.

  • A rational number can be written as a fraction (e.g., \(\frac{1}{2}, -5, 0.7\)).
  • An irrational number cannot be written exactly as a fraction or a terminating/repeating decimal (e.g., \(\pi\), or \(\sqrt{2}\)).

If you try to calculate \(\sqrt{2}\) on your calculator, you get $1.41421356...$ This number goes on forever without repeating.

When a question asks for an exact value, it means you must leave your answer in surd form (or in terms of $\pi$).

Surd or Not a Surd?

A square root is a surd only if the number under the root sign is not a perfect square.

  • Surds: \(\sqrt{3}, \sqrt{7}, \sqrt{15}\)
  • Not Surds: \(\sqrt{9} = 3\), \(\sqrt{16} = 4\), \(\sqrt{0.25} = 0.5\)
Quick Review:

We use surds to provide exact answers, avoiding rounding errors from messy decimals.

1. Simplifying Surds (E1.17.1)

Just like you simplify a fraction (e.g., \(\frac{10}{20} = \frac{1}{2}\)), you must always simplify a surd to its simplest form.

The goal of simplifying is to move any perfect square factors out from under the root sign.

Step-by-Step Simplification

Prerequisite: Remember your first few square numbers: 4, 9, 16, 25, 36, 49, 64, 81, 100...

  1. Find the largest perfect square factor: Look for the largest square number that divides exactly into the number under the root.
  2. Separate the roots: Rewrite the surd as the multiplication of two separate roots, one containing the perfect square.
  3. Simplify the perfect square root: Calculate the square root of the perfect square factor.
Example: Simplify \(\sqrt{20}\)

(This is an example specifically mentioned in the syllabus notes!)

1. The perfect square factors of 20 are 1 and 4. The largest is 4.
2. Separate the roots: \[\sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5}\] 3. Simplify: \[\sqrt{4} \times \sqrt{5} = 2\sqrt{5}\]

So, \(\sqrt{20}\) simplifies to \(\mathbf{2\sqrt{5}}\).

Analogy: Think of the number under the root (the radicand) as a hotel. Only perfect squares can check out! 4 is a perfect square, so it gets to leave, but it checks out as its square root, 2. The remaining factor (5) stays inside the root.

The Key Rules for Operating Surds

We use these rules constantly for multiplication and division:


Rule 1 (Multiplication): \[\sqrt{a} \times \sqrt{b} = \sqrt{ab}\] Example: \(\sqrt{3} \times \sqrt{5} = \sqrt{15}\)
Example: \(\sqrt{4} \times \sqrt{9} = 2 \times 3 = 6\). Also \(\sqrt{4 \times 9} = \sqrt{36} = 6\).

Rule 2 (Division): \[\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\] Example: \(\frac{\sqrt{18}}{\sqrt{2}} = \sqrt{\frac{18}{2}} = \sqrt{9} = 3\)

2. Adding and Subtracting Surds (Combining Like Terms)

You can only add or subtract surds if they have the same value inside the square root. These are called like surds.

Think of the surd \(\sqrt{a}\) as an unknown variable, like \(x\). You can only combine terms that have the same variable.

  • \(3\sqrt{2} + 5\sqrt{2} = 8\sqrt{2}\) (Like adding \(3x + 5x = 8x\)).
  • \(7\sqrt{3} - \sqrt{3} = 6\sqrt{3}\) (Remember $\sqrt{3}$ is \(1\sqrt{3}\)).
  • \(4\sqrt{5} + 2\sqrt{7}\) cannot be combined.

Solving Combined Expressions

Sometimes, surds don't look like they can be combined, but they can be simplified into like surds first.

Example: Work out \(\sqrt{200} - \sqrt{32}\)

(This is an example specifically mentioned in the syllabus notes!)

1. Simplify \(\sqrt{200}\): The largest square factor of 200 is 100.
\[\sqrt{200} = \sqrt{100 \times 2} = 10\sqrt{2}\] 2. Simplify \(\sqrt{32}\): The largest square factor of 32 is 16.
\[\sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2}\] 3. Subtract the like surds:
\[\sqrt{200} - \sqrt{32} = 10\sqrt{2} - 4\sqrt{2} = \mathbf{6\sqrt{2}}\]

Common Mistake Alert!

You CANNOT add numbers under the root sign: \(\sqrt{9} + \sqrt{16} = 3 + 4 = 7\). But \(\sqrt{9+16} = \sqrt{25} = 5\). Since \(7 \neq 5\), the rule is simple: Do not add or subtract the numbers inside the surds. Simplify first!

3. Rationalising the Denominator (E1.17.2)

In mathematics, it is considered bad form (or often, incomplete) to leave a surd in the denominator of a fraction. Rationalising the denominator is the process of eliminating the surd from the bottom of the fraction, usually by multiplying by 1 (in a special form).

Type A: Single Surd in the Denominator

If the denominator is just \(\sqrt{a}\), multiply the top and bottom of the fraction by \(\sqrt{a}\). Remember that \(\sqrt{a} \times \sqrt{a} = a\).

Example: Rationalise \(\frac{10}{\sqrt{5}}\)

(This is an example specifically mentioned in the syllabus notes!)

  1. Multiply the fraction by \(\frac{\sqrt{5}}{\sqrt{5}}\): \[\frac{10}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}}\]
  2. Calculate the numerator and denominator: \[\frac{10 \times \sqrt{5}}{\sqrt{5} \times \sqrt{5}} = \frac{10\sqrt{5}}{5}\]
  3. Simplify the fraction (the rational parts): \[\frac{10\sqrt{5}}{5} = \mathbf{2\sqrt{5}}\]

Type B: Binomial Denominator (Using Conjugates)

This is the Extended content step that often trips students up. If the denominator looks like \(a + \sqrt{b}\) or \(\sqrt{a} - \sqrt{b}\), we cannot simply multiply by the surd part. We must use its conjugate.

The conjugate is formed by simply changing the sign between the two terms.

  • If the denominator is \(3 + \sqrt{2}\), the conjugate is \(\mathbf{3 - \sqrt{2}}\).
  • If the denominator is \(\sqrt{5} - 1\), the conjugate is \(\mathbf{\sqrt{5} + 1}\).

Why use the conjugate? It uses the difference of two squares formula: \[(a+b)(a-b) = a^2 - b^2\] When applied to surds, this makes the square roots disappear! \[(a+\sqrt{b})(a-\sqrt{b}) = a^2 - (\sqrt{b})^2 = a^2 - b \quad \text{(No more surd!)}\]

Example: Rationalise \(\frac{1}{-1+\sqrt{3}}\)

(This is an example specifically mentioned in the syllabus notes, written slightly differently to match the algebraic order \(\sqrt{3} - 1\)).

1. Rewrite the denominator for clarity: \(\frac{1}{\sqrt{3} - 1}\).
2. The conjugate of \(\sqrt{3} - 1\) is \(\mathbf{\sqrt{3} + 1}\).
3. Multiply the fraction by the conjugate over itself: \[\frac{1}{\sqrt{3} - 1} \times \frac{\sqrt{3} + 1}{\sqrt{3} + 1}\]
4. Calculate the denominator (using the difference of squares formula): \[(\sqrt{3} - 1)(\sqrt{3} + 1) = (\sqrt{3})^2 - (1)^2 = 3 - 1 = 2\]
5. Calculate the numerator: \[1 \times (\sqrt{3} + 1) = \sqrt{3} + 1\]
6. Write the final rationalised fraction: \[\frac{\sqrt{3} + 1}{2} \quad \text{or} \quad \mathbf{\frac{1+\sqrt{3}}{2}}\]

Did you know? Rationalising the denominator was historically very important because dividing by an irrational number used to be impossible to calculate manually! Calculators handle it easily today, but we still learn it to provide mathematically correct and simplified forms.

Key Takeaways for Surds

  • Always aim to simplify surds by factoring out perfect squares (e.g., \(\sqrt{50} = 5\sqrt{2}\)).
  • Only like surds (those with the same root part) can be added or subtracted.
  • To rationalise a denominator with a single surd \(\sqrt{a}\), multiply by \(\frac{\sqrt{a}}{\sqrt{a}}\).
  • To rationalise a denominator with a binomial surd (\(a \pm \sqrt{b}\)), multiply by its conjugate.

Keep practicing these steps—you've got this!