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Surds are irrational numbers that are roots of rational number. This document breaks down Surd and gives everything you need to know about Surd.
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SURDSDEFINITION OF SURDS...............................................................................................................................1
RULES FOR SIMPLIFYING SURDS................................................................................................................1
BASIC SURDS..............................................................................................................................................2
ADDITION AND SUBTRACTION OF SURDS..................................................................................................4
MULTIPLICATION OF SURDS......................................................................................................................5
CONJUGATE SURDS....................................................................................................................................6
RATIONALIZATION OF SURDS....................................................................................................................8
APPLICATION OF SURDS...........................................................................................................................10
EXERSCISE.................................................................................................................................................13
SUMMARY................................................................................................................................................15
DEFINITION OF SURDS
A surd is an irrational number that is actually the root of
a rational number. An irrational number is an unending
decimal fraction that can only be approximated to a
precise value like pi ( π ). Examples of surds include: √ 2, √ 3,
√ 5, √ 6 , √7 , √ 10 √ 11, e.t.c.
RULES FOR SIMPLIFYING SURDS
There are two main rules used for simplifying surds.
These rules are:
Product Rule: √ x x √ y = √ x x y
= √ xy
1
,Quotient Rule: √√ y =
x
√ x
y
Note: There is no addition and subtraction rules for
surds.
That is,
√x + √ y ≠ √ x+ y
Also,
√x - √ y ≠ √x− y
BASIC SURDS
Basic surds are surds that have been reduced to their
lowest forms. For example, √ 8 can be written in its basic
form as 2√ 2.
To arrive at basic surds, we just need to split the rational
number in the root into the product of two numbers,
such that one is a perfect square.
In the case of √ 8, the rational number is 8 and can be
written in the form of 4 x 2. You see here that 4 is the
perfect square.
2
, This means that, √ 8 = √ 4 x 2
= √4 x √2
= 2 x √2
= 2√ 2
Example: Change each of the following to their basic
forms:
i. √ 12
ii. √ 24
iii. √ 45
Solution
i. √ 12 = √4 x 3
= √4 x √3
= 2 x √3
= 2√ 3
ii. √ 24 = √4 x 6
= √4 x √6
= 2 x √6
= 2√ 6
iii. √ 45 = √9 x5
= √9 x √5
= 3 x √5
= 3√ 5
3
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