# Thread: rationalize expressions

1. ## rationalize expressions

hi i was wondering if any one can help me rationalize these four expressions and if possible show working out so i can understand better, thanks in advance.

1)
8_
√2

2)
16 √9
√12 3√8

3
√2 (1+√2)

4)
1+
√2
1- √2

2. 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.