Question

if a = 7+4√3
find a + 1 /a​

Answer 1

Given:—

• $\large\bold{\mathtt{\color{red}{★ a = \:7 \:+ \:4√3}}}$

To find :—

• $\large\bold{\mathtt{\color{red}{→ \:a \:+\:\frac{1}{a}}}}$

Now,

$\large\bold{a + \frac{1}{a} }$

$\large \bold{ = \frac{ {a}^{2} + 1}{a} }$

$\large \bold{= \frac{ {(7 + 4 \sqrt{3} )}^{2} + 1}{7 + 4 \sqrt{3} } }$

$\large\bold{ = \frac{49 + 56 \sqrt{3} + 16 \times 3 }{7 + 4 \sqrt{3} } }$

$\large \bold {= \frac{49 + 56 \sqrt{3} + 48}{7 + 4 \sqrt{3} } }$

$\large \bold{ = \frac{97 + 56 \sqrt{3} }{7 + 4 \sqrt{3} } }$

${ \underline \frak{ \bold{‡ Now, \: we \: will \: \: rationalize \: \: the \: \: denominator‡}}}$

$\large \bold{ :→ \frac{(97 + 56 \sqrt{3})(7 - 4 \sqrt{3} ) }{(7 + 4 \sqrt{3})(7 - 4 \sqrt{3}) } }$

$\bold{ = \frac{97 \times 7 - 97 \times 4 \sqrt{3} + 56 \sqrt{3} \times 7 - 56 \sqrt{3} \times 4 \sqrt{3} }{49 - 16 \times 3} }$

$= \frac{679 - 388 \sqrt{3} + 392 \sqrt{3} - 672 }{49 - 48}$

$\bold{ = 679 - 672 + 392 \sqrt{3} - 388 \sqrt{3} }$

$\large \underbrace { \color{orange}{\bold{:→ 7 + 4 \sqrt{3} }}}$

$\Large{\colorbox{orange}{\underline{\underline{♪Answer♪:—\:\:7+4√3} }}}$