Question

If √5=2•236 and √3=1•732 find the value of 2\√5+√3+7\√5-√3

Answer 1

Identity used :

$(x + y)(x - y) = {x}^{2} - {y}^{2}$


Given,

√5 = 2.236
√3 = 1.732

Now,

$\frac{2}{ \sqrt{5} + \sqrt{3} } + \frac{7}{ \sqrt{5} - \sqrt{3} }$

On rationalizing the denominator we get,

$= \frac{2}{ \sqrt{5} + \sqrt{3} } \times \frac{ \sqrt{5} - \sqrt{3} }{ \sqrt{5} - \sqrt{3} } + \frac{7}{ \sqrt{5} - \sqrt{3} } \times \frac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5} + \sqrt{3} } \\ \\ = \frac{2( \sqrt{5} - \sqrt{3} )}{ {( \sqrt{5} )}^{2} - {( \sqrt{3} )}^{2} } + \frac{7( \sqrt{5} + \sqrt{3} ) }{ {( \sqrt{5} )}^{2} - {( \sqrt{3} )}^{2} } \\ \\ = \frac{2 \sqrt{5} - 2 \sqrt{3} }{5 - 3} + \frac{7 \sqrt{5} + 7 \sqrt{3} }{5 - 3} \\ \\ = \frac{2 \sqrt{5} - 2 \sqrt{3} }{2} + \frac{7 \sqrt{5} + 7 \sqrt{3} }{2} \\ \\ = \frac{2 \sqrt{5} - 2 \sqrt{3} + 7 \sqrt{5} + 7 \sqrt{3} }{2} \\ \\ = \frac{9 \sqrt{5} + 5 \sqrt{3} }{2} \\ \\ = \frac{9 \times 2.236 + 5 \times 1.732}{2} \\ \\ = \frac{20.124 + 8.66}{2} \\ \\ = \frac{28.784}{2} \\ \\ = 14.392$