SURDS

When the root of a number cannot be determined exactly, it is referred to as a Surds. The answers to Surds are normally irrational numbers. For instance, are referred to as surds meanwhile numbers such as are not.
 

Examples of Surds

Examples of perfect squares

Surds are great way to handle irrational numbers without the aid of a calculator that is why we study them in mathematics.

There are two laws which are useful when doing calculations with surds:

Multiplication law: 

Division law:      

Examples (Multiplication and Division laws)

From the above examples the following conclusions can be deduced:

These conclusions are important as they simplify calculations as we will see later on the post.

ADDITION AND SUBTRACTION OF SURDS

When adding and subtracting surds it is important to note that the root is treated as a “variable” and the number in front of the surd as the “coefficient”, similar to addition and subtraction of like terms in expressions.

Examples (Addition and Subtraction of Surds)

SURDS WITH NON-PRIME BASES

In case you have not noticed but examples used in the previous discussions we only used prime bases. Surds with non-prime bases can be computed, this is where factors and perfect squares join our discussions.

Examples:

Write the following in surd form and simplify without using a calculator

*In this example we see that 20 was expressed as a product of two of its factors 4 x 5 with one of the factors a prefect square (Multiplication law reverse engineered). This is how you should simplify surds with non-prime bases.

SIMPLIFICATION OF SURDS

In a surd expression I recommend you start off by simplifying all surds with non-prime bases, then apply the properties for surds discussed where applicable (multiplication, division, subtraction and addition laws).

Examples:

Written by Elliot Mashiane