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, Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number. is not a surd, as the answer is a whole number. But Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number. is not a whole number. You could use a calculator to find that Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number. 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  Basic rules  A surd is a square root which cannot be reduced to a whole number.

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

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 Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number. , although occasionally it can be Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number. .

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:

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

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

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

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

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

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

Note:

Always simplify surds before adding or subtracting them.

 

Example 9

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

Solution:

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

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

Now  Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number.  = 3 and  Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number.  = 2,

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

= 3 + 2

=  5

However  Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number.  +  Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number.  is not  Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number.  =  Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number.  = 3.60555

Hence remember  Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number.  +  Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number.   Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number.    Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number. , and

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

 

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

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

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

=  2 Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number.   +  3 Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number.

=  5 Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number.

Here we are adding the two surds only when they are alike, i.e. both the surds have  Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number. , so we could add them together – exactly like how it is done in Algebra – adding like terms.

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

=  3 Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number.   –  2 Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number.

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

 

Examples of addition and subtraction of surds

  1. Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number.   +  5 Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number.   –  3 Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number.

=   Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number.  (2 + 5 – 3)

=  4 Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number.

  1. Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number.   +  2 Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number.   –  6 Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number.

=   Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number.  (1 + 2 – 6)

=  -3 Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number.

  1. Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number.   +  5 Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number.   –  2 Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number.   –   Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number.

=  (3 Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number.   –  2 Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number. )  +  (5 Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number.   –   Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number. )

=   Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number.   +  4 Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number.

  1. Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number.   +  2 Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number.   –   Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number.

=   Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number.   +  2 Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number.   –   Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number.

=  3 Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number.   +  4 Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number.   –  6 Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number.

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

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

=  3 Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number.   +   Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number.   +   Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number.

=  9 Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number.   +  2 Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number.   +  4 Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number.

=  11 Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number.   +  4 Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number.

Year 9 Interactive Maths – Second Edition

Multiplication of Surds

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

In general:

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

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

Example 10

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

Solution:

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

Short way:

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

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

Short way:

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

 

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

In general:

The multiplication of unlike surds gives an irrational number.

Example 11

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

Solution:

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

Note:

Always give your answer in the simplest possible form.

 

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

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

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

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

In general:

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

Example 12

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

Solution:

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

Alternatively:

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

The Distributive Law

Surds  Basic rules  A surd is a square root which cannot be reduced to a whole number.
Example 13

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

Solution:

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

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

Setting out:

Often, we set out the solution as follows:

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

 

Example 14

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

Solution:

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

Now let us consider the use of the following formulas:

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

Example 15

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

Solution:

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

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

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

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