1. Number (Edexcel GCSE Maths)

Revision Note

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 square root of blank end 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. square root of 1 equals 1, square root of 4 equals 2, square root of 9 equals 3, square root of 16 equals 4).

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


Examples of surds:

  • square root of 2 space almost equal to space 1.41
  • square root of 15 space almost equal to space 3.87
  • 2 square root of 3 space almost equal to space 2 space cross times space 1.73 space almost equal to space 3.46


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 square root of 72 space

  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. square root of 72 space equals square root of 36 space cross times space 2 end root space
  3. Simplify the square root of the square
    1. square root of 36 space cross times 2 end root space equals square root of 36 space cross times square root of 2 space equals 6 square root of 2 space

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: square root of 3 cross times square root of 5 space end root equals square root of 3 cross times 5 end root equals square root of 15 space
  2. Dividing: fraction numerator square root of 18 over denominator square root of 3 end fraction equals square root of 18 over 3 end root equals square root of 6

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: 3 square root of 2 plus 4 square root of 2 equals 7 square root of 2
  2. Subtraction: 5 square root of 7 minus 2 square root of 7 equals 3 square root of 7

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 3 plus square root of 5we 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: 

fraction numerator 2 over denominator 3 plus square root of 5 end fraction

To rationalise the denominator in this case, we'll multiply both the numerator and the denominator by the conjugate of the denominator, which is 3 minus square root of 5.

fraction numerator 2 over denominator 3 plus square root of 5 end fraction cross times fraction numerator 3 minus square root of 5 over denominator 3 minus square root of 5 end fraction

Now, multiply the numerators together and the denominators together:

fraction numerator 2 cross times open parentheses 3 minus square root of 5 close parentheses over denominator open parentheses 3 plus square root of 5 close parentheses cross times open parentheses 3 minus square root of 5 close parentheses end fraction equals fraction numerator 6 minus 2 square root of 5 over denominator 9 minus 5 end fraction equals fraction numerator 6 minus 2 square root of 5 over denominator 4 end fraction

So, the rationalised fraction is:

fraction numerator 6 minus 2 square root of 5 over denominator 4 end fraction

Surds in Quadratic Equations

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

x equals fraction numerator negative b plus-or-minus square root of b squared minus 4 a c end root over denominator 2 a end fraction

b squared minus 4 a c 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 x squared minus 4 x plus 2 equals 0.

By applying the quadratic formula, we get:

a equals 1 comma space b equals negative 4 comma space c equals 2

x equals fraction numerator negative open parentheses negative 4 close parentheses plus-or-minus square root of open parentheses negative 4 close parentheses squared minus 4 cross times 1 cross times 2 end root over denominator 2 cross times 1 end fraction
x equals fraction numerator 4 plus-or-minus square root of 16 minus 8 end root over denominator 2 end fraction
x equals fraction numerator 4 plus-or-minus square root of 8 over denominator 2 end fraction

Now, we need to simplify the surd:

square root of 8 equals square root of 4 cross times 2 end root equals 2 square root of 2

So, the roots of the equation are:

x equals fraction numerator 4 plus-or-minus 2 square root of 2 over denominator 2 end fraction
x equals 2 plus-or-minus square root of 2


Surds Exam Tip

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