To rationalize is to make the denominator a rational number. Meaning, do not leave a surd or radical number in the denominator.

That's my understanding, so:

8 /sqrt(2)

Multiply both numerator and denominator by sqrt(2),

= [8sqrt(2)] /2

= 4sqrt(2) --------------------answer.

[16sqrt(9)] /[sqrt(12) *3sqrt(8)]

There many ways to do this.

One way is to simplify first all of the radicals.

= [16*3] /[sqrt(4*3) *3sqrt(4*2)]

= 48 /[2sqrt(3) *6sqrt(2)]

= 48 /[12sqrt(3*2)

= 4 /sqrt(6)

Multiply both numerator and denominator by sqrt(6),

= 4sqrt(6) /6

= (2/3)sqrt(6) ---------------answer.

sqrt(2)*[1 +sqrt(2)]

= [sqrt(2) *1 ] +[sqrt(2) *sqrt(2)]

= sqrt(2) +2 ------------------------------answer.

[1 +sqrt(2)] /[1 -sqrt(2)]

That denominator needs to be multiplied by its conjugate [1 +sqrt(2)]---which just happens to be the numerator here---to make the denominator a difference of two squares.

Multiply both numerator and denominator by [1 +sqrt(2)],

= [1 +sqrt(2)]^2 /[1 -2]

= [1 +2sqrt(2) +2] /[-1]

= -[3 +2sqrt(2)]

= -3 -2sqrt(2) -------------------answer.