Surds Basic rules A surd is a square root which cannot be reduced to a whole number.

Ss 3 mathematic surds

Surds

Basic rules

A surd is a square root which cannot be reduced to a whole number. For example,  is not a surd, as the answer is a whole number. But  is not a whole number. You could use a calculator to find that  but instead of this we often leave our answers in the square root form, as a surd.

You need to be able to simplify expressions involving surds. Here are some general rules that you will need to learn.

Surds

Surds are numbers left in ‘square root form’ (or ‘cube root form’ etc). They are therefore irrational numbers. The reason we leave them as surds is because in decimal form they would go on forever and so this is a very clumsy way of writing them.

Surds

Surds are irrational numbers left in their root form. Most of the time this is in the form  , although occasionally it can be  .

The reason for this is that it is much easier to write numbers in this form, as opposed to decimals that continue forever.

Examples of numbers written in surd forms

A few more examples:

As 35 is not a square number, this cannot be simplified any more.

You may notice that we can simplify the last answer even more.

Addition of surds

Only like surds can be added or subtracted. Example 8 Solution:

Note: Always simplify surds before adding or subtracting them.

Example 9

Solution:

Addition and Subtraction of Surds

Addition and subtraction of surds involve a few simple rules:

1. we can add or subtract surds only when they are in the simplest form, and
2. we can add or subtract like surds only.

Let’s look at both these rules one by one. First lets consider    +  

Now    = 3 and    = 2,

so    +  

= 3 + 2

=  5

However    +    is not    =    = 3.60555

Hence remember    +          , and

  –         

Now the second rule states that we can add or subtract like surds only. So lets consider

   +   

=      +   

=  2    +  3 

=  5 

Here we are adding the two surds only when they are alike, i.e. both the surds have   , so we could add them together – exactly like how it is done in Algebra – adding like terms.

Similarly     –   

=  3    –  2 

=   

Examples of addition and subtraction of surds

1. 2    +  5    –  3 

=     (2 + 5 – 3)

=  4 

1.    +  2    –  6 

=     (1 + 2 – 6)

=  -3 

1. 3    +  5    –  2    –   

=  (3    –  2  )  +  (5    –    )

=      +  4 

1.    +  2    –   

=      +  2    –   

=  3    +  4    –  6 

=   

1. 3    +      +   

=  3    +      +   

=  9    +  2    +  4 

=  11    +  4 

In general: The multiplication of like surds gives a rational number. That is:

Example 10 Solution: Short way: Short way:

In general: The multiplication of unlike surds gives an irrational number. Example 11 Solution: Note: Always give your answer in the simplest possible form.

We can get the same result by multiplying the radicands and multiplying the coefficients, as follows: In general:

Example 12 Solution: Alternatively:

The Distributive Law Example 13 Solution: Setting out: Often, we set out the solution as follows:   Example 14 Solution: Now let us consider the use of the following formulas: Example 15 Solution: