Question

express 21/3√5+√3 with rational denominator​

Answer 1

$\textbf{Given:}$

$\mathsf{\dfrac{21}{3\sqrt{5}+\sqrt{3}}}$

$\textbf{To express:}$

$\mathsf{\dfrac{21}{3\sqrt{5}+\sqrt{3}}\;with\;rational\;denominator}$

$\textbf{Solution:}$

$\mathsf{Here,\;we\;have\;to\;rationalise\;the\;denominator\;of\;\dfrac{21}{3\sqrt{5}-\sqrt{3}}}$

$\mathsf{To\;rationalize\;the\;denominator\;multiply\;both\;numerator}$

$\mathsf{and\;denominator\;by\;3\sqrt{5}-\sqrt{3}}$

$\mathsf{\dfrac{21}{3\sqrt{5}+\sqrt{3}}{\times}\dfrac{3\sqrt{5}-\sqrt{3}}{3\sqrt{5}-\sqrt{3}}}$

$\mathsf{=\dfrac{21(3\sqrt{5}-\sqrt{3})}{(3\sqrt{5}+\sqrt{3})(3\sqrt{5}-\sqrt{3})}}$

$\mathsf{Using\;identity\bf\;(a-b)(a+b)=a^2-b^2}$

$\mathsf{=\dfrac{21(3\sqrt{5}-\sqrt{3})}{3^2(\sqrt{5})^2-(\sqrt{3})^2}}$

$\mathsf{=\dfrac{21(3\sqrt{5}-\sqrt{3})}{9(5)-3}}$

$\mathsf{=\dfrac{21(3\sqrt{5}-\sqrt{3})}{45-3}}$

$\mathsf{=\dfrac{21(3\sqrt{5}-\sqrt{3})}{42}}$

$\mathsf{=\dfrac{3\sqrt{5}-\sqrt{3}}{2}}\;\textsf{which is having rational denominator}$

$\implies\boxed{\mathsf{\dfrac{21}{3\sqrt{5}+\sqrt{3}}=\dfrac{3\sqrt{5}-\sqrt{3}}{2}}}$

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