When the denominator of a fraction is of the form **to/ ^{n}**

**√b**or

**to/(**

^{n}**√b +**

^{n}**√c)**we must make the necessary transformations so that the radicals of the denominator disappear, this process is called

**denominator rationalization**.

Rationalizing the denominator consists of converting a fraction whose denominator is irrational, for example √7, into an equivalent whose denominator is rational.

**Rationalize the denominator of a fraction when it is a monomial**

We multiply the numerator and denominator by a **radical of the same index as the denominator, whose subradical part contains the same letters and coefficients of said radical raised to exponents, such that, added to the ones it already has, it gives us the index or a multiple of it**.

**Example 1**: Rationalize the denominator of the following expression: x / ^{6}√x^{4}.

- We multiply the numerator and denominator by
^{6}√. - The subradical part will be x
^{2}, since x^{4}∙ x^{2 }= x^{(4+2) }= x^{6}(which is the same number as the index of the radical).

Thus:

x / ^{6}√x^{4 }= (x / ^{6}√x^{4 }) · ( ^{6}√x^{2} / ^{6}√x^{2} ) = (x ·^{6}√x^{2 }) / ^{6}√x^{4} = (x ·^{6}√x^{2}) / x = ^{6}√x^{2}

Let us observe that the result is a rational quantity.

**Example 2**: Rationalize the denominator of the following expression: ∛ (4x^{2} / y∛x)

∛ (4x^{2} / y∛x) = ∛ (4x^{2}) / ∛ (y∛x) = ∛ (4x^{2}) / ∛ [∛ (xy^{3})] = ∛ (4x^{2}) / ^{9}√ (xy^{3})

We multiply the numerator and denominator by ^{9}√. The subradical part will be x^{8}y^{6}, since (x^{1}∙ and^{3}) ∙ (x^{8}∙ and^{6}) = x^{(1+8)}∙ and^{(3+6) }= x^{9}∙ and^{9} (which are the same numbers in the index of the radical).

Thus:

∛ (4x^{2}) / ^{6}√ (xy^{3}) = ∛ (4x^{2}) / ^{9}√ (xy^{3}) ∙ [^{9}√ (x^{8}y^{6}) / ^{9}√ (x^{8}y^{6}) ] =

^{= 9}√ [(4x^{2})^{3}x^{8}y^{6}] / ^{9}√ (x^{9}y^{9}) =

= ^{9}√ (64x^{14}y^{6}) / (xy) =

= x · ^{9}√ (64x^{5}y^{6}) / (xy) =

^{= 9}√ (64x^{5}y^{6}) / Y

## Rationalize the denominator of a fraction when it is a binomial

We multiply the numerator and denominator by the **conjugated expression of the denominator**, that is, if the denominator is a sum, we multiply by its difference, if it is a subtraction, by its sum:

- to / (
^{n}√b +^{n}√c) = [a / (^{n}√b +^{n}√c)] · [(^{n}√b -^{n}√c) / (^{n}√b -^{n}√c)] - to / (
^{n}√b -^{n}√c) = [a / (^{n}√b -^{n}√c)] · [(^{n}√b +^{n}√c) / (^{n}√b +^{n}√c)]

**Example 1**: Rationalize the denominator of the following expression: 9 / (√2 + √11)

We multiply the numerator and denominator by the conjugated expression of the denominator, that is, √2** – **√11

9 / (√2 + √11) = [9 / (√2 + √11)] · [(√2 - √11) / (√2 - √11)]

Remembering that (a + b) (a - b) = a^{2 }- b^{2} you have:

9 / (√2 + √11) = [9 (√2 - √11)] / [(√2)^{2 }- (√11)^{2} ] =

= [9 (√2 - √11)] / (2 - 11) =

= [9 (√2 - √11)] / - 9 =

= √2 - √11

**Example 2**: Rationalize the denominator of the following expression √3 / (√5 - √2)

We multiply the numerator and denominator by the conjugated expression of the denominator, that is, √5 + √2

√3 / (√5 - √2) = [√3 / (√5 - √2)] · [(√5 + √2) / (√5 + √2)] =

= [√3 (√5 + √2)] / [(√5)^{2} - (√2)^{2} ] =

= (√3√5 + √3√2) / (5 - 2) =

= (√15 + √6) / 3

E**example 3**: Rationalize the denominator of the following expression: ^{4}√ [5^{4 }/ (3 - √2)]

We multiply the numerator and denominator by the conjugated expression of the denominator, that is, 3 + √2

^{4}√ [5^{4 }/ (3 - √2)] = ^{4}√ {[5^{4 }/ (3 - √2)] · [(3 + √2) / (3 + √2)]} =

^{ = 4}√ {[5^{4} (3 + √2)] / [(3)^{2} - (√2)^{2} ] } =

^{ = 4}√ {[5^{4} (3 + √2)] / [9 - 2]} =

^{ = 4}√ {[5^{4} (3 + √2)] / 5} =

^{ = 4}√ [5^{4} (3 + √2)] / ^{4}√5

Now, we multiply the numerator and denominator by ^{4}√ The subradical part will be 5^{3}, since 5^{1}5^{3 }= 5^{1+3 }= 5^{4} (which is the same number as the index of the radical).

^{4}√ [5^{4} (3 + √2)] / ^{4}√5 = {^{4}√ [5^{4} (3 + √2)] / ^{4}√5} · (^{4}√5^{3 }/ ^{4}√5^{3}) =

^{ = 4}√ [5^{7} (3 + √2)] / ^{4}√5^{4} =

^{ = 4}√ 5^{3 }(3 + √2)

Thus,

^{4}√ [5^{4 }/ (3 - √2)] = ^{4}√ [125 (3 + √2)]