Surds

At the end 0f this tutorial students ought to be able state what surd means, perform the mathematical operations of addition, subtraction, multiplication and division of surds. The students would as well be able to rationalize denominators of surds.

Definition

When you studied numbers, you were told that a number that is "square" is one that can be expressed as the square of a few other rational numbers. For instance:

9 = 32

81 = 92

(4/9) = (2/3)2

However, not all numbers are rational numbers, that is they do not possess exact square roots. Examples are:

√2

√3

√5

√8

2√3 and so on.

The square roots of numbers that do not possess exact square roots are called Surds.

Numbers like that are referred to as irrational numbers.

Even though estimated square roots of irrational numbers can be found from tables of square roots, it is normally simpler to work with SURDS themselves. You must however not that at this stage whenever you work with square roots, only positive square roots are taken into consideration.

Reduction to Basic Form

Any surd which is made up of a square number as a factor inside the radical - the square root sign is not in the basic form. For instance

√2

√50

√108 , are not in the basic form since they could be reduced further through simplification. The following examples will showcase this concept:

√27=√ (9X3) = √9√3 = 3√3

√50 = √ (25 X 2) =√ 25 √2 = 5√2

√108 = √ (36 X 3) =√ 36 √3 = 6√3

From the three examples above, 3√3, 5√2 and 6√3 cannot be simplified further; they have been reduced to their basic form. Surds that cannot be simplified any further are said to be in their basic form.

Addition and Subtraction of Similar Surds

Surds that are in the basic form can be added and subtracted. The example below will demonstrate the idea.

Simplify √80 +√20 -√45

Solution:

First reduce all the surds to their basic forms.

That is,√80+√20-√45

= √(16X5) + √(4X5) - √(9X5) RHS(the right hand side):

=4√ 5 + 2√5 - 3√5 = 6√5 - 3√5 = 3√5

Observe that the above surds can be added or subtracted because they are in comparable form, that is, numbers under the radical signs are the same and they have the same index. Mixed surds like 2√2 + 2√7 - 2√3 are not equivalent, therefore, they cannot be added or subtracted. This means that they cannot be simplified further.

Multiplication and Division of Surds

Multiplication and division of surds are performed through two fundamental laws of surds.

i) For multiplication of surds, the rule is: √a √b = √(a.b)

Examples 1:

√5 √3

= √(5X3 )

= √(15)

Example 2

√ 2 √7

= √ (2X7)

= √ (14)

For division of Surd, the following rule applies: √a /√b =√ (a/b)

Examples: 3

√6/√3 = & radical (6/3)

=√2

Example 4

√18/√3 = √(18/3)

= √6

Rationalizing the Denominator

A surd like √3/5 cannot be simplified further; but a surd like 2/√3 can be written in a convenient form, since it is not normal to have the radical ie the square root symbol in the denominator. The process of removing the radical from the denominator is known as rationalization. In order to perform rationalization, you must have knowledge of conjugate surds.

Conjugate Surds:

Given a surd (a - √b), its conjugate is defined as (a + √b) and vice-versa.

When a surd is multiplied by its conjugate, their product is no more a surd.

For instance:

Multiply (a+√b) by its conjugate (a-√b) to obtain (a+√b)(a-√b)

= a2 - a√b+a√b-√b√b

= a2 - √b√b

=a2-b2

You can see that the result is no longer a surd. Now, the rule is: To rationalize a radical denominator of a surd, multiply both numerator and denominator of the surd by the conjugate of the denominator.

Points to remember about Surds

A surd is a square root which cannot be reduced to a whole number. For example,√4 = 2 is not a surd. But √5 that is not a whole number is surd. You could use a calculator to find that but, instead of this; we frequently leave our answers in the square root form, as a surd.

Surds are mathematical expressions that contain square roots. Nevertheless, it must be emphasized that the square roots are 'irrational' i.e. they do not result in a whole number, a terminating decimal or a recurring decimal. View more on psalmfresh.blogspot.com for more educating and tricks article Thank you