Rationalising factor is a term with which a term is multiplied or divided to make the whole term rational.

**Rationalisation of Surds**

Rationalising
factor is a term with which a term is multiplied or divided to make the whole term
rational.

Examples:

(i) √3 is a rationalising factor of √3 (since √3 × √3
= the rational number 3)

(ii) ^{7}√5^{4} is a rationalising factor
of ^{7}√5^{3} (since their product = ^{7}√5^{7}
= 5 , a rational)

**Thinking Corner**

1. In the example (i) above, can √12 also be a rationalising factor? Can you think of any other number
as a rationalising factor for √3
?

2. Can you think of any other number as a rationalising factor for
^{7}√5^{3} in example
(ii) ?

3. If there can be many rationalising factors for an expression containing
a surd, is there any advantage in choosing the smallest among them for manipulation?

**Progress Check**

Identify a rationalising factor for each one of the following surds
and verify the same in each case:

(i) √18 (ii) 5√12 (iii) ^{3}√49 (iv) 1/√8

Can you guess a rationalising factor for 3 + √2 ? This
surd has one rational part and one radical part. In such cases, the rationalising
factor has an interesting form.

A rationalising factor for 3 + √2 is 3 − √2 . You can
very easily check this.

(3+ √2)(3− √2) = 3^{2}− (√2)^{2}

= 9 −2

= 7, a rational.

What could be the rationalising factor for a +* *√*b
*where a and* b *are rational numbers? Is it a − √*b *? Check it. What
could be the rationalising factor for √*a* +* *√*b *where a and*
b *are rational numbers? Is it √*a* − √*b *? Or, is it − √*a*
+* *√*b *? Investigate.

Surds
like *a* + √*b* and
*a* − √*b* are called
conjugate surds. What is the conjugate of √*b *+*a
*? It is* *−*
*√*b *+*a
*. You would have perhaps noted by now
that a conjugate is usually obtained by changing the sign in front of the surd!

**Example 2.27**

Rationalise
the denominator of

*Solution*

(i)
Multiply both numerator and denominator by the rationalising factor √14

Tags : Conjugate Surds | Example Solved Problems | Real Numbers | Maths , 9th Maths : UNIT 2 : Real Numbers

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9th Maths : UNIT 2 : Real Numbers : Rationalisation of Surds | Conjugate Surds | Example Solved Problems | Real Numbers | Maths

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