Question

Simplify:
`(1)/(√(5)-√(7)+√(12))`

Answer 1

Multiply the numerator and denominator by the conjugate.

Exact Form:

1√5−√7+2√3

Decimal Form:

0.75267290…

Answer 2

Make use of the difference of squares identity,

`a^2-b^2=(a-b)(a+b)`

Let
`a=\sqrt5-\sqrt7` and
`b=√(12)`. Then

`\frac1{(\sqrt5-\sqrt7)+√(12)}\cdot((\sqrt5-\sqrt7)-√(12))/((\sqrt5-\sqrt7)-√(12))=((\sqrt5-\sqrt7)-√(12))/((\sqrt5-\sqrt7)^2-(√(12))^2)=(\sqrt5-\sqrt7-√(12))/(5-2√(35)+7-12)=-(\sqrt5-\sqrt7-√(12))/(2√(35))`

Now multiply the numerator and denominator by √(35):

`-(\sqrt5-\sqrt7-√(12))/(2√(35))\cdot(√(35))/(√(35))=-((\sqrt5-\sqrt7-√(12))√(35))/(70)=-(5\sqrt7-7\sqrt5-2√(105))/(70)`