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Rationalization

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In mathematics, rationalization is a process that is undertaken to make the fraction denominator real especially in the cases, where we may find surds or complex numbers in the denominator. Dividing any number by a complex number is not likely to be possible, as we do not know exact value of imaginary number. In case, we are having surds in the fraction, although it is possible for us to find the fraction value without rationalization, calculations become tough as compared to division when done using real numbers.

Rationalization is explained as a process of simplifying an expression by eliminating the radicals without changing the value of the expression.

Rationalize the Denominator

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Rationalizing the denominator is a process of eliminating the roots in the fractions.

There are different methods to rationalize a denominator as demonstarted below:

Case 1: Where there is only a monomial denominator (Single square root).
In this case, multiply both the numerator and the denominator by whatever makes the denominator an expression so that, it contains no longer a radical. If it simplifies the expression, you can even multiply the numerator and denominator by it.

Case 2: If the denominator is of the type a + $\sqrt{b}$, then multiply and divide with the conjugate and then, apply the formula.
(a - b)(a + b) = (a$^{2}$ - b$^{2}$) for the denominator. Then, the irrational denominator becomes rational. This is the process to rationalize the denominator of this kind.

Rationalization Examples

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Given below are some of the examples on rationalization.

Solved Examples

Question 1: Simplify $\frac{2}{\sqrt{5}}$
Solution:
$\frac{2}{\sqrt{5}}$

= $\frac{2}{\sqrt{5}}$ $\times$ $\frac{\sqrt{5}}{\sqrt{5}}$

= $\frac{2\sqrt{5}}{5}$

Question 2: Simplify $\frac{3}{5-\sqrt{2}}$
Solution:
$\frac{3}{5-\sqrt{2}} \times \frac{5+\sqrt{2}}{5+\sqrt{2}}$

= $\frac{3(5+\sqrt{2})}{5^2 - (\sqrt{2})^2}$

= $\frac{3(5+\sqrt{2})}{23}$

Rationalize the Numerator

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Rationalizing numerator is similar to rationalizing the denominator. In this method, we multiply numerator and denominator by a radical that will get rid of the radical in the numerator.

Solved Examples

Question 1: Solve $\sqrt{\frac{8}{9}}$ 
Solution:
Rationalize the numerator $\sqrt{\frac{8}{9}}$

= $\frac{\sqrt{8}}{\sqrt{9}} \times \frac{\sqrt{8}}{\sqrt{8}}$

= $\frac{\sqrt{64}}{\sqrt{72}}$

= $\frac{8}{\sqrt{72}}$

Question 2: Solve $\frac{\sqrt{x}-\sqrt{3}}{3-x}$
Solution:
$\frac{\sqrt{x}-\sqrt{3}}{3-x}$ = $\frac{\sqrt{x}-\sqrt{3}}{3-x} \times \frac{\sqrt{x}+\sqrt{3}}{\sqrt{x}+\sqrt{3}}$ (Rationalize the numerator)

$\frac{x-3}{(3-x)(\sqrt{x}+\sqrt{3})}$

= $\frac{-1}{(\sqrt{x}+\sqrt{3})}$