Question

Rationale the denominator and simplify 5+2√3/7+4√3

Answer 1

$\frac{5 + 2 \sqrt{3} }{7 + 4 \sqrt{3} } \\ = \frac{(5 + 2 \sqrt{3})(7 - 4 \sqrt{3} ) }{(7 + 4 \sqrt{3} )(7 - 4 \sqrt{3}) } \\ = \frac{35 - 20 \sqrt{3} + 14 \sqrt{3} - 24 }{ {(7)}^{2} - {(4 \sqrt{3}) }^{2} } \\ = \frac{11 - 6 \sqrt{3} }{49 - 48} = 11 - 6 \sqrt{3}$

Answer 2

$\Huge{\underline{\underline{\blue{\mathfrak{Answer :}}}}}$

To Find :

We have to rationalize

$\sf{ \frac{5 + 2 \sqrt{3} }{7 + 4 \sqrt{3} } }$

Solution :

We know that,

When we rationalize a number then we multiply denominator by numertaor and both denominator by changing the last sign.

Now,

$\rule{200}{2}$

$\sf{ \implies\frac{5 + 2 \sqrt{3} }{7 + 4 \sqrt{3} } \times \frac{7 - 4 \sqrt{3} }{7 - 4 \sqrt{3} } } \\ \\ { \LARGE{ \leadsto{ \boxed{ \boxed{ \green{ \sf{{a}^{2} - {b}^{2} = (a + b)(a - b) }}}}}}} \\ \\ \bf{\hookrightarrow {Putting \: values}}\\ \\ \sf{ \implies\frac{5 + 2 \sqrt{3}(7 - 4 \sqrt{3})}{ {(7)}^{2} - {( 4 \sqrt{3} )}^{2} } } \\ \\ \sf{ \implies\frac{35 - 20 \sqrt{3} + 14 \sqrt{3} - 24}{49 - 48} } \\ \\ \sf{ \implies\frac{11 - 6 \sqrt{3} }{1} } \\ \\ { \LARGE{ \leadsto{ \boxed{ \boxed{ \blue{ \sf{11 - 6 \sqrt{3} }}}}}}}$