Surds

Surds can seem like a tricky topic to grasp, but fear not! These tips from the experts at Save My Exams will help you understand and master surds, empowering you to tackle any surds-related questions in your exams.

What are Surds?

Surds are irrational numbers that can't be simplified into a neat fraction. In other words, they're numbers that contain a square root   where the number inside the square root isn't a square number. Squares are numbers, like 1, 4, 9, and 16, have whole numbers as their square roots (e.g. , , , ).

Surds are usually written in the form  or  where a is a positive integer that isn't a perfect square and b is any real number.

Examples of surds:

• 
• 
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Simplifying Surds

Sometimes, we can simplify surds to make them easier to work with. Here's a step-by-step guide to help you simplify surds using the example of 

1. Identify the largest perfect square that is a factor of the given surd.
   1. The largest square factor of 72 is 36
2. Write the number within the square root as a product of the square and another number
   1. 
3. Simplify the square root of the square
   1. 

Multiplying and Dividing Surds

Multiplying and dividing surds are straightforward operations. When multiplying, you can multiply the numbers within square roots and keep them under the same square root symbol. When dividing, divide the numbers within square roots and keep them under the same square root symbol.

Examples:

1. Multiplying:
2. Dividing:

Adding and Subtracting Surds

Adding and subtracting surds can be done when the numbers within the square roots are the same. In this case, treat the surds like algebraic terms.

Examples:

1. Addition:
2. Subtraction:

Rationalising the Denominator

Sometimes, you'll encounter a fraction with a surd in the denominator (the bottom of the fraction). In these cases, you'll need to rationalise the denominator, this just means removing the surd from the denominator.

To rationalise the denominator, multiply both the numerator and the denominator by the denominator. If there is an expression which forms the denominator such as we just need to change the sign in between the terms. This expression with the opposite sign is called the conjugate.

Multiplying the top and bottom of the fraction by the same thing is the equivalent of multiplying by 1, this means the overall value doesn't change, it just allows us to write the fraction in a different form.

Example: 

To rationalise the denominator in this case, we'll multiply both the numerator and the denominator by the conjugate of the denominator, which is .

Now, multiply the numerators together and the denominators together:

So, the rationalised fraction is:

Surds in Quadratic Equations

Surds may also appear in quadratic equations. When solving these equations, you may need to use the quadratic formula:

is the discriminant, this determines the nature of the roots for any quadratic. If the discriminant is positive and not a perfect square, the roots of the quadratic equation will be surds.

Example: Solve the quadratic equation .

By applying the quadratic formula, we get:

Now, we need to simplify the surd:

So, the roots of the equation are:

Surds Exam Tip

Remember to simplify surds whenever possible to make expressions and equations easier to handle.