Rationalization


 
 
Concept Explanation
 

Rationalization

RATIONALIZATION:

Let a and be positive real numbers.Then,

large (i)(sqrt{a}+sqrt{b})(sqrt{a}-sqrt{b})=a-b

large (ii)(a+sqrt{b})(a-sqrt{b})=a^{2}-b

large (iii)(sqrt{a}pm sqrt{b})^{2}=apm 2sqrt{ab}+b

large (iv)(sqrt{a}+sqrt{b})(sqrt{c}+sqrt{d})=sqrt{ac}+sqrt{ad}+sqrt{bc}+sqrt{bd}

Rationalisation of Denominator:

Sometimes we come across expressions containing square roots in their denominators. Addition, subtraction, multiplication and division of such expressions in convenient if their denominators are free from square roots.To make the denominators free from square roots, we multiply the numerator and denominator by an irrational number. Such a number is called rationalisation factor.

Consider the expression large frac{3+sqrt{5}}{sqrt{2}}.

We know that large sqrt{2}timessqrt{2}=2. Therefore, to remove the square root from the denominator we multiply its numerator and denominator by large sqrt{2}.

large therefore      large frac{3+sqrt{5}}{sqrt{2}}=frac{3+sqrt{5}}{sqrt{2}timessqrt{2}}=sqrt{2}=frac{3sqrt{2}+sqrt{5}timessqrt{2}}{2}=frac{3sqrt{2}+sqrt{10}}{2}

Let us now consider the expression large frac{1}{2+sqrt{3}}

We know that large (a+sqrt{b})(a-sqrt{b})=a^{2}-b

large therefore       large (2+sqrt{3})(2-sqrt{3})=4-3=1, which is a rational number

So, we multiply the numerator and denominator by large 2-sqrt{3}

large therefore ;frac{1}{2+sqrt{3}}=frac{1}{2+sqrt{3}}timesfrac{2-sqrt{3}}{2-sqrt{3}}=frac{2-sqrt{3}}{4-3}=2-sqrt{3}

Example: If large x=2+sqrt{3}, find the valur of large x^{2}+frac{1}{x^{3}}

SOLUTION  We have, large x=2+sqrt{3},

large therefore       large frac{1}{x}=frac{1}{2+sqrt{3}}=frac{1}{2+sqrt{3}}timesfrac{2-sqrt{3}}{2-sqrt{3}}=frac{2-sqrt{3}}{2^{2}-(sqrt{3})^{2}}=frac{2-sqrt{3}}{4-3}=2-sqrt{3}

Now,   large x^{2}+frac{1}{x^{2}}=left ( x+frac{1}{x} right )^{2}-2

large Rightarrow ;;x^{2}+frac{1}{x^{2}}=(2+sqrt{3}+2-sqrt{3})^{2}-2=4^{2}-2=16-2=14

 

Sample Questions
(More Questions for each concept available in Login)
Question : 1

If sqrt{13-asqrt{10}} =sqrt8+sqrt5 ,then a=

Right Option : C
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Explanation
Question : 2

On rationalzing the denominator,   large frac{1}{sqrt{5}+sqrt{2}}   we get

Right Option : C
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Explanation
Question : 3

If   large x=sqrt5+2 , then x-frac{1}{x} equals

Right Option : B
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Explanation
 
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