Help with multiplying and adding radicals?

a radical is just the "√" sign what really matters, is the index outside the radical.
standard radicals (ones with no index shown) are assumed to be square roots so I will work as if that's what you are dealing with.

You cannot just add radicals.
sqrt(x)+sqrt(y)≠ sqrt(x+y)
you just can't add them so the simplest form is
sqrt(x)+sqrt(y)

when you multiply radicals with the same index, you multiply the numbers inside the radical
sqrt(x) * sqrt(y) = sqrt(xy)

so now that we have the basics down, lets look at your equation.

sqrt(14) *[sqrt(2) + sqrt(7)]
well you cant add the ones inside the brackets so you just have to multiply the sqrt(14) through

= [sqrt(14)*sqrt(2) + sqrt(14)*sqrt(7)]
=sqrt(2*14) + sqrt(7*14)
=sqrt(28) + sqrt(98)

if you want to reduce your radicals, you can, or stop here if your teacher allows.

=sqrt(2*2*7) + sqrt(7*7*2)
=2*sqrt(7) + 7*sqrt(2) ,;,;,;,; Final Answer

For non-negative a and b: sqrt(a)sqrt(b) = sqrt(ab)
Use the distributive law:
sqrt(14)(sqrt2 + sqrt7) = sqrt(14)sqrt(2) + sqrt(14)sqrt(7)

If I write sqrt(14) as sqrt(2)sqrt(7)

= sqrt(2)sqrt(7)sqrt(2) + sqrt(2)sqrt(7)sqrt(7)

= 2sqrt(7) + 7sqrt(2)

which is fully simplified

You can probably solve this after reading the info in the following website.
http://www.themathpage.com/alg/simplify-radicals.htm