Question

if a =2+√3 find the value of a³+1/a³​

Answer 1

$\huge\sf\red{\underline{\underline{Given}}}\::$

$\begin{cases}\sf\gray{a \ = \ 2 \ + \ \sqrt{3} }\end{cases}$

$\huge\sf\blue{\underline{\underline{To\:Find}}}\::$

$\begin{cases}\sf\gray{The\ value\ of \ : \ a^3+\dfrac{1}{a^3} }\end{cases}$

$\huge\sf\pink{\underline{\underline{Solution}}}\::$

$\sf\underline\orange{Firstly \ let \ us \ calculate \ a^3}$

${\sf{\orange{a^3=(2+\sqrt{3} )^3}}}\\\\ \mapsto\:\:{\sf{\purple{ a^3=(2)^3 + (\sqrt{3} )^3 + 3 \times 2 \times \sqrt{3} (2+\sqrt{3} )}}}\\\\\mapsto\:\:{\sf{\green{ a^3=8+3\sqrt{3}+6\sqrt{3}(2+\sqrt{3} ) }}}\\\\\mapsto\:\:{\sf{\purple{ a^3=8+3\sqrt{3}+(6\sqrt{3}\times 2)+(6\sqrt{3}\times\sqrt{3} ) }}}\\\\\mapsto\:\:{\sf{\green{ a^3=8+3\sqrt{3} +12\sqrt{3} +18}}}\\\\\mapsto\:\:{\sf{\purple{ a^3=26+15\sqrt{3} }}}$

$\sf\orange{Now \ calculating \ \dfrac{1}{a^3}}$

$\displaystyle\mapsto\:\:{\sf{\purple{ \frac{1}{a^3}=\frac{1}{26+15\sqrt{3}} }}}\\\\\displaystyle\mapsto\:\:{\sf{\green{ \frac{1}{a^3}=\frac{26-15\sqrt{3}}{(26+15\sqrt{3})(26-15\sqrt{3})} }}}\\\\\displaystyle\mapsto\:\:{\sf{\purple{ \frac{1}{a^3}= \frac{26-15\sqrt{3}}{(26)^2-(15\sqrt{3})^2} }}}\\\\\displaystyle\mapsto\:\:{\sf{\green{ \frac{1}{a^3}= \frac{26-15\sqrt{3}}{676-675} }}}\\\\\displaystyle\mapsto\:\:{\sf{\purple{ \frac{1}{a^3}= \frac{26-15\sqrt{3}}{1} }}}$

$\displaystyle\mapsto\:\:{\sf{\green{ \frac{1}{a^3}= 26-15\sqrt{3} }}}$

$\mapsto\:\:{\sf{\purple{ a+ \frac{1}{a^3}=26+15\sqrt{3}+(26-15\sqrt{3}) }}} \\\\ \mapsto\:\:{\sf{\green{ a+ \frac{1}{a^3}=26+15\sqrt{3}+26-15\sqrt{3}}}} \\\\ \mapsto\:\:{\boxed{\sf{\pink{ a+ \frac{1}{a^3}=52}}}}$