Sometimes when you have expressions with radicals on the denominator, you'd like to change it into an expression where there are no radicals in the denominator.
(When you write radical 7, I'll write sqrt(7) for "square root of 7").
So if you had 1/sqrt(7), sometimes, you want to rewrite this without the root in the bottom. To do that, you'd multiply numerator and denominator by sqrt(7).
Ultimately, you're multiplying by 1, but by multiplying the denominator by sqrt(7), we are able to get rid of the root.
So you'd end up with [sqrt(7)] / [sqrt(7)*sqrt(7)] = sqrt(7) / 7.
If the denominator is an expression like 5 - sqrt(7),
you want to multiply it by "The Conjugate",
which is 5 + sqrt(7).
The conjugate flips the sign of the sqrt term and keeps the sign of the term without the sqrt.
So the conjugate of 1 + sqrt(2) is 1 - sqrt(2).
The conjugate of 1 - sqrt(2) is 1 + sqrt(2).
The conjugate of -1 - sqrt(2) is -1 + sqrt(2).
Anyways, by multiplying by the conjugate, you'd get rid of the root,
because (5 + sqrt(7))(5 - sqrt(7)) = 25 - 7 (after you FOIL it out).
This is true because (A + B)(A - B) = A^2 - B^2.
And so by multiplying the numerator and denominator of the fraction you have by 5 + sqrt(7), you'd be able to rewrite the fraction without roots in the denominator.