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Rationalization

Number System - Concepts

Natural Number Whole number and Negative Integers

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Rationalization

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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 a=2+ $sqrt3$ and b=2- $sqrt3$ then $frac{1}{a^2}-frac{1}{b^2}$ is equal to
A. 14		B. -14
C. 8 $sqrt3$		D. -8 $sqrt3$
Right Option : D
View Explanation

Question : 2

If $large x=frac{sqrt5+sqrt3}{sqrt5-sqrt3};;and;; y=frac{sqrt5-sqrt3}{sqrt5+sqrt3}$ ,then x + y + xy =
A. 9	B. 5	C. 17	D. 7
Right Option : A
View Explanation

Question : 3

If $large x=sqrt5+2$ , then $x-frac{1}{x}$ equals
A. $2sqrt5$	B. 4	C. 2	D. $sqrt5$
Right Option : B
View Explanation

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