Question

rationalise the denominator ​

Answer 1

Step-by-step explanation:

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Answer 2

★ Rationalised form of – 1/(√3 + √2) is – (√3 – √2).

Step-by-step explanation

Rationalisation is defined as the process converting the irrational denominator of a fraction to a rational number by multiplying both the numerator and denominator with the rationalising factor.

As per the given question, we have been asked to rationalise the denominator of:

$\twoheadrightarrow\quad\underline{\boxed{ \bf{ \dfrac{ - 1}{ \sqrt{3} + \sqrt{2} } }}}$

To rationalise, first we have to find the rationalising factor of the given fraction. Rationalising factor of an irrational term is simply got by changing the sign to its respective opposite one.

Here, the denominator is given by (√3 + √2). So, its rationalising factor is (√3 – √2). Now, multiply this rationalising term with both the numerator and the denominator.

$\twoheadrightarrow\quad{ \sf{ \dfrac{ - 1}{ \sqrt{3} + \sqrt{2} } } \times \dfrac{ \sqrt{3} - \sqrt{2} }{ \sqrt{3} - \sqrt{2} } }$

Rearranging the terms using brackets.

$\twoheadrightarrow\quad{ \sf{ \dfrac{ - 1( \sqrt{3} - \sqrt{2} )}{( \sqrt{3} + \sqrt{2} )( \sqrt{3} - \sqrt{2} )} }}$

Simplify the numerator by multiplying (– 1) with it.

$\twoheadrightarrow\quad{ \sf{ \dfrac{ - ( \sqrt{3} - \sqrt{2} )}{( \sqrt{3} + \sqrt{2} )( \sqrt{3} - \sqrt{2} )} }}$

The denominator is in the form of algebraic identity: (a + b)(a – b) = a² – b². The denominator can be simplified using this identity where, a = √3 and b = √2.

$\twoheadrightarrow\quad{ \sf{ \dfrac{ - ( \sqrt{3} - \sqrt{2} )}{ {( \sqrt{3} )}^{2} - {( \sqrt{2}) }^{2} } }}$

Further simplifying the denominator by squaring the roots.

$\twoheadrightarrow\quad{ \sf{ \dfrac{ - ( \sqrt{3} - \sqrt{2} )}{ (\sqrt{3} \times \sqrt{3}) - ( \sqrt{2} \times \sqrt{2}) } }}$

Writing the values of squares of the numericals.

$\twoheadrightarrow\quad{ \sf{ \dfrac{ - ( \sqrt{3} - \sqrt{2} )}{3 - 2 } }}$

Subtracting the numbers in the denominator.

$\twoheadrightarrow\quad{ \sf{ \dfrac{ - ( \sqrt{3} - \sqrt{2} )}{1} }}$

This can also be written as:

$\twoheadrightarrow\quad{ \sf{ - ( \sqrt{3} - \sqrt{2} )}}$

Now, look at the denominator. It is a rational number.

∴ The denominator has been rationalised.