For the first two, start by factoring everything as much as you can. That will give you some binomials to factor out, leaving the remaining problem much simpler.

(3) All this stuff is multiplied together. Remember that sqrt(a^2 * b) = a*sqrt(b). Find perfect squares and give each one its own sqrt sign, and then take the root.

(4) 27 and 12 are multiples of 3. Factor out the 3 (still under the sqrt), and then treat each term as above. Textbook authors generall make it easy on you (ha!) at this point, in that they’ll likely leave you with three terms whose only radical is sqrt(3).

(5) Treat sqrt(6) and sqrt(2) as variables. If it helps, replace them with x and y at first. Do the multiplications using the binomial multiplication law (FOIL, if you learned that). Then put the sqrt terms back in and finish the algebra. Note that sqrt(2)*sqrt(6) = sqrt (2*6), which you simplified in #4.

(6) Multiply by the conjugate of the denominator. If you have a denominator of (a+b), and one or both are square roots, then multiply through by (a-b)/(a-b). the denominator of (a+b)(a-b) will simplify nicely as the difference of two perfect squares.