Question

rationalise the denominator of 1 upon 2 + root 3

Answer 1

$\underline{\textbf{Given:}}$

$\mathsf{\dfrac{1}{2+\sqrt{3}}}$

$\underline{\textbf{To rationalize:}}$

$\mathsf{The\;denominator\;of\;\dfrac{1}{2+\sqrt{3}}}$

$\underline{\textbf{Solution:}}$

$\underline{\textbf{Concept used:}}$

$\mathsf{Conjugate\;of\;a+\sqrt{b}\;is\;a-\sqrt{b}}$

$\mathsf{Consider,}$

$\mathsf{\dfrac{1}{2+\sqrt{3}}}$

$\textsf{Multiply both numerator and denominator by}\;\mathsf{2-\sqrt{3}}$

$\mathsf{=\dfrac{1}{2+\sqrt{3}}{\times}\dfrac{2-\sqrt{3}}{2-\sqrt{3}}}$

$\textsf{Using the identity,}$

$\boxed{\mathsf{(a-b)(a+b)=a^2-b^2}}$

$\mathsf{=\dfrac{2-\sqrt{3}}{2^2-(\sqrt{3})^2}}$

$\mathsf{=\dfrac{2-\sqrt{3}}{4-3}}$

$\mathsf{=\dfrac{2-\sqrt{3}}{1}}$

$\mathsf{=2-\sqrt{3}}$

$\underline{\textbf{Answer:}}$

$\boxed{\mathsf{\dfrac{1}{2+\sqrt{3}}=2-\sqrt{3}}}$

Answer 2

Step-by-step explanation:

Hope this help you.

The answer of the question is 2-√3