Question

simplify: 2/square root of 2 + square root of 5

Answer 1

\[\frac{ 2 }{ \sqrt2 } + \frac{ \sqrt5 }{ 1 }=\frac{ 2 }{ \sqrt2 } + \frac{\sqrt2 \sqrt5 }{ \sqrt2 }=\frac{ 2+\sqrt10 }{ \sqrt2 }\]

But we can't have a radical in the denominator so we multiply the numerator and denominator by square root of 3. This would be look like the following:
\[\frac{ \sqrt3(2+\sqrt10) }{ \sqrt3\sqrt3 }=\frac{2\sqrt3+\sqrt30) }{ 3 }\]

You can leave the solution in this form which is known as the exact solution.

Answer 2

Sorry, I accidentally multiplied the top and bottom by square root of 3 when I was supposed to multiply it by the square root of 2. The correct solution would look like this:
\[\frac{ \sqrt2(2+\sqrt10) }{ \sqrt2\sqrt2 }=\frac{ 2\sqrt2+\sqrt20 }{2 }=\frac{ 2\sqrt2+2\sqrt5 }{2 }=\frac{ 2(\sqrt2+\sqrt5) }{2} = \sqrt2+\sqrt5\]