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. |

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. Given below are some of the examples on rationalization.

### Solved Examples

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

**Solution:**

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

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

**Question 2:**Simplify $\frac{3}{5-\sqrt{2}}$

**Solution:**

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

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

### Solved Examples

**Question 1:**Solve $\sqrt{\frac{8}{9}}$

**Solution:**

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

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

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

**Question 2:**Solve $\frac{\sqrt{x}-\sqrt{3}}{3-x}$

**Solution:**

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

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