How do you simplify (√2+√5) /( √2-√5)?

Question

27604 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-03-16T22:59:50

The simplified expression is $-(7+2sqrt10)/3$ .

$color(white)=(sqrt2+sqrt5)/(sqrt2-sqrt5)$

$=((sqrt2+sqrt5))/((sqrt2-sqrt5))color(red)(*((sqrt2+sqrt5))/((sqrt2+sqrt5)))$

$=((sqrt2+sqrt5)(sqrt2+sqrt5))/((sqrt2-sqrt5)(sqrt2+sqrt5))$

$=((sqrt2+sqrt5)(sqrt2+sqrt5))/(sqrt2^2+sqrt2sqrt5-sqrt2sqrt5-sqrt5^2)$

$=((sqrt2+sqrt5)(sqrt2+sqrt5))/(2color(red)cancelcolor(black)(+sqrt10-sqrt10)-5)$

$=((sqrt2+sqrt5)(sqrt2+sqrt5))/(2-5)$

$=((sqrt2+sqrt5)(sqrt2+sqrt5))/(-3)$

$=(sqrt2^2+sqrt2sqrt5+sqrt2sqrt5+sqrt5^2)/(-3)$

$=(2+sqrt10+sqrt10+5)/(-3)$

$=(2+2sqrt10+5)/(-3)$

$=(7+2sqrt10)/(-3)$

$=-(7+2sqrt10)/3$

This is the answer. You can verify using a calculator:

![https://www.desmos.com/calculator]( useruploads.socratic.org )