Question

rationalise the denominator of 1/2+root3
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Answer 1

Given -

• $\sf \: \dfrac{1}{2 + \sqrt{3} }$

To -

• Rationalise the denominator

Identity used -

• 3rd Algerbric identity = (a + b) (a - b) = a² - b²

Solution -

In the question, we are given with a fraction, and we need to rationalise the denominator, for that, we will multiply the denominator with both numerator and denominator, remember that we will change the sign of denominator when we will multiply so that the algebraic identity can be used and then we will do the square of denominator, then we will subract the denominator that will give us the required answer. Let's do it!

$\sf \longrightarrow \: \dfrac{1}{2 + \sqrt{3} } \: \times \dfrac{2 - \sqrt{3} }{2 - \sqrt{3} } \\ \\ \\ \sf \longrightarrow \: \dfrac{2 - \sqrt{3}}{ {(2)}^{2} - {( \sqrt{3})}^{2}} \: (3rd \: identity \: used) \\ \\ \\ \sf \longrightarrow \: \dfrac{2 - \sqrt{3} }{4 \: - \: 3} \\ \\ \\ \sf \longrightarrow \: \dfrac{2 - \sqrt{3} }{1} \\ \\ \sf \: or \: \\ \\ \sf \longrightarrow \: 2 - \sqrt{3} \: \: \: ans$

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Answer 2

Step-by-step explanation:

$\large\bf{\underline{\underline{We\ have:-}}}$

$\bf : \implies {\dfrac{1}{2\ +\ \sqrt{3}}}$

$\large\bf{\underline{\underline{Solution:-}}}$

❏ By using third identity:-

• (a + b) (a - b) = a² - b².

$\bf : \implies {\dfrac{1}{2\ +\ \sqrt{3}}} \\ \\ \\ \bf : \implies {\dfrac{1}{2\ +\ \sqrt{3}}\ \times\ \dfrac{2\ -\ \sqrt{3}}{2\ -\ \sqrt{3}}} \\ \\ \\ \bf : \implies {\dfrac{2\ -\ \sqrt{3}}{2^2\ -\ (\sqrt{3})^2}} \\ \\ \\ \bf : \implies {\dfrac{2\ -\ \sqrt{3}}{4\ -\ 3}} \\ \\ \\ \bf : \implies {\dfrac{2\ -\ \sqrt{3}}{1}} \\ \\ \\ \bf : \implies {\underline{\boxed{\purple{\bf 2\ -\ \sqrt{3}}}}}\bigstar$

‎

$\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\qquad\qquad\boxed{\bf{\mid{\overline{\underline{\pink{\bigstar More\ to\ know \bigstar}}}}}\mid}\\\\\end{gathered}\end{gathered}\end{gathered}\end{gathered}$

‎

❏ Identity I :- (a + b)² = a² + 2ab + b².

❏ Identity II :- (a – b)² = a² - 2ab + b².

❏ Identity III :- (a + b) (a - b) = a² - b².

❏ Identity IV :- (x + a) (x + b) = x² + (a + b) x + ab.

❏ Identity V :- (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca.

❏ Identity VI :- (a + b)³ = a³ + b³ + 3ab (a + b).

❏ Identity VII :- (a - b)³ = a³ - b³ - 3ab (a – b).

❏ Identity VIII :- a³ + b³ + c³ - 3abc = (a + b + c) (a² + b² + c² - ab - bc - ca).