Question

find a and b √7+√10 upon √7-√10=a+b√70​

Answer 1

given :- (√7 + √10)/(√7 - √10) = a + b√70

$\sf LHS : - \\ \\ \tt = \frac{ \sqrt{7} + \sqrt{10} }{ \sqrt{7 } - \sqrt{10} } \\ \\ \tt = \frac{ \sqrt{7} + \sqrt{10} }{ \sqrt{7} - \sqrt{10} } \times \frac{ \sqrt{7} + \sqrt{10} }{ \sqrt{7} + \sqrt{10} } \\ \\ \tt = \frac{ {( \sqrt{7} + \sqrt{10} )}^{2} }{( \sqrt{7} - \sqrt{10} )( \sqrt{7} + \sqrt{10} )} \\ \\ \tt = \frac{ {( \sqrt{7} )}^{2} + 2( \sqrt{7})( \sqrt{10} ) + ( { \sqrt{10} })^{2} }{ {( \sqrt{7} )}^{2} - ( { \sqrt{10} })^{2} } \\ \\ \tt = \frac{7 + 2 \sqrt{70} + 10}{7 -10 } \\ \\ \tt = \frac{17 + 2 \sqrt{70} }{ - 3} \\ \\ \tt = \frac{ - 17 - 2 \sqrt{70} }{3}$

RHS :- a + b√70

comparing LHS with RHS :-

➡ (-17 - 2√70)/3 = a + b√3

• a = -17/3

and

• b = -2/3

identities used :-

⏺ (a + b)² = a² + 2ab + b²

⏺ (a + b) (a - b) = a² - b²