Introduction to Surds

Have you ever noticed that if you type \(\sqrt{2}\) into your calculator, the decimal digits go on forever without repeating? This is because \(\sqrt{2}\) is an irrational number. Writing out a long decimal is messy and inaccurate, so in Mathematics (Specification B), we keep these numbers in their root form to stay 100% accurate. These are what we call Surds.

In this chapter, you will learn how to "tidy up" these numbers by simplifying them and how to move a surd from the bottom of a fraction to the top—a process called rationalising the denominator. Don't worry if it seems tricky at first; once you spot the patterns, it becomes as easy as basic algebra!

Note: For more on how surds fit into the wider world of numbers, see the chapter on "Fractions, decimals, ratio, proportion and percentage".

1. What is a Surd?

A surd is the square root of a number that is not a perfect square. For example:

  • \(\sqrt{9} = 3\) (This is not a surd because 9 is a perfect square).
  • \(\sqrt{2}\) is a surd because 2 is not a perfect square.
  • \(\sqrt{5}\) is a surd.

Did you know? Using surds allows us to provide exact answers. If a question asks for an "exact value," you should usually leave your answer in surd form rather than using a calculator to find a decimal.

2. Simplifying Surds

Simplifying a surd means making the number inside the square root as small as possible. To do this, we look for square factors (numbers like 4, 9, 16, 25, etc.) that go into the number.

The Rule:

\(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\)

Step-by-Step Example: Simplify \(\sqrt{48}\)
  1. Find the largest square number that is a factor of 48. In this case, it is 16 (\(16 \times 3 = 48\)).
  2. Write the surd as a product: \(\sqrt{16 \times 3}\).
  3. Split the surd: \(\sqrt{16} \times \sqrt{3}\).
  4. Square root the perfect square: \(4 \times \sqrt{3}\).
  5. The final simplified answer is \(4\sqrt{3}\).

Quick Tip: If you can't find the largest square factor right away, don't panic! You can do it in smaller steps. For example: \(\sqrt{48} = \sqrt{4 \times 12} = 2\sqrt{12}\). Then, simplify \(\sqrt{12}\) into \(\sqrt{4 \times 3}\) to get \(2 \times 2\sqrt{3} = 4\sqrt{3}\).

3. Adding and Subtracting Surds

You can only add or subtract surds if they are "like terms." This means the number under the square root must be the same.

The Rule: Treat the surd like an \(x\) in algebra.

  • \(5\sqrt{3} + 2\sqrt{3} = 7\sqrt{3}\)
  • \(10\sqrt{5} - 4\sqrt{5} = 6\sqrt{5}\)
  • \(\sqrt{2} + \sqrt{3}\) cannot be simplified further!

Common Mistake: Many students try to add the numbers inside the roots, like \(\sqrt{2} + \sqrt{3} = \sqrt{5}\). This is incorrect! Always keep the roots separate unless you are multiplying or dividing.

4. Multiplying and Dividing Surds

When multiplying or dividing, you can combine the numbers under one square root sign.

Multiplication:

\(\sqrt{a} \times \sqrt{b} = \sqrt{ab}\)

Example: \(\sqrt{2} \times \sqrt{5} = \sqrt{10}\)

Division:

\(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\)

Example: \(\frac{\sqrt{10}}{\sqrt{2}} = \sqrt{5}\)

A Special Trick:

When you multiply a surd by itself, the root disappears: \(\sqrt{a} \times \sqrt{a} = a\).
For example, \(\sqrt{5} \times \sqrt{5} = 5\).

5. Rationalising the Denominator

In Mathematics, it is considered "untidy" to have a surd on the bottom of a fraction (the denominator). To fix this, we rationalise it.

Type 1: Simple fractions

If the denominator is just a single surd, multiply the top and the bottom of the fraction by that surd.

Example: Rationalise \(10 \times \frac{1}{\sqrt{5}}\)

  1. Combine into one fraction: \(\frac{10}{\sqrt{5}}\).
  2. Multiply the top and bottom by \(\sqrt{5}\): \(\frac{10 \times \sqrt{5}}{\sqrt{5} \times \sqrt{5}}\).
  3. This becomes \(\frac{10\sqrt{5}}{5}\).
  4. Simplify the fraction: \(10 \div 5 = 2\), so the final answer is \(2\sqrt{5}\).

Type 2: Denominators with two terms

If the denominator looks like \(\sqrt{a} - b\) or \(\sqrt{a} + b\), you must multiply the top and bottom by its "partner" (the same terms but with the opposite sign).

Example: Rationalise \(\frac{15}{\sqrt{7} - 2}\)

  1. The denominator is \(\sqrt{7} - 2\), so our "partner" is \(\sqrt{7} + 2\).
  2. Multiply top and bottom: \(\frac{15(\sqrt{7} + 2)}{(\sqrt{7} - 2)(\sqrt{7} + 2)}\).
  3. Expand the bottom: \((\sqrt{7} \times \sqrt{7}) + (2\sqrt{7}) - (2\sqrt{7}) - (2 \times 2)\).
  4. Notice how the middle terms cancel out: \(7 - 4 = 3\).
  5. Now the fraction is \(\frac{15(\sqrt{7} + 2)}{3}\).
  6. Simplify: \(15 \div 3 = 5\), so the final answer is \(5(\sqrt{7} + 2)\) or \(5\sqrt{7} + 10\).

Key Takeaways Summary

Quick Review:

  • Simplify by finding the largest square factor: \(\sqrt{48} = 4\sqrt{3}\).
  • Add/Subtract like terms only: \(5\sqrt{3} + 2\sqrt{3} = 7\sqrt{3}\).
  • Multiply/Divide numbers under the root: \(\sqrt{2} \times \sqrt{3} = \sqrt{6}\).
  • Rationalise by multiplying the top and bottom to remove the root from the denominator.
  • Use opposite signs for complex rationalising: for \((\sqrt{a} - b)\), multiply by \((\sqrt{a} + b)\).

Don't forget: you can always use your calculator to check your work! While you must show your steps to get full marks in Paper 1 and Paper 2, your calculator's \(\sqrt{x}\) button will usually show the simplified surd or rationalised form automatically.