Question 202704
{{{3sqrt(2) + 5sqrt( 2  )}}}
I get {{{(3+5) sqrt( 2 ) = 8 sqrt(2)}}}

This is exactly right. The only reason you are allowed to add {{{3sqrt(2) + 5sqrt( 2  )}}} is that the square roots are the same. It is very much like adding 3x and 5x and getting 8x. We add them using the Distributive Property (in reverse). This is also the reason the second problem is incorrect. The square roots are different.

The correct way to simplify
{{{3sqrt(8) + 2sqrt(16)}}}
is to start by simplifying each square root. The {{{sqrt(16)}}} is simply 4. The {{{sqrt(8)}}} is not so easy. We start by trying to find perfect square factors in 8. We should find that 8 = 4*2 (with 4 being the perfect square, of course). So {{{sqrt(8) = sqrt(4*2)}}}. Now we can use one of the basic properties of square roots: {{{sqrt(a*b) = sqrt(a) * sqrt(b)}}} to split our square root into the product of square roots: {{{sqrt(4*2) = sqrt(4)*sqrt(2)}}}. Now that {{{sqrt(4)}}} is separate we can replace it with 2 giving {{{sqrt(8) = sqrt(4*2) = sqrt(4) * sqrt(2) = 2*sqrt(2)}}}<br>
Putting this all together we get:
{{{3*sqrt(8) + 2sqrt(16) = 3*2*sqrt(2) + 2*4 = 6*sqrt(2) + 8}}}
and we can go no further. {{{6*sqrt(2)}}} and {{{8}}} are unlike terms in much the same way that 6x and 8 are unlike terms. You just cannot add them. So we are finished.