Question

How do you rationalize this? (move irrationals out of the denominator)

Answer 1

Just do it in two steps

1 + root2 + root3 = (1 + root2) + root3

Answer 2

Can’t figure out what to multiply the top and bottom by to get the irrationals out of the bottom :/

Answer 3

(1+sqrt(2)+sqrt(3))(1-sqrt(2)-sqrt(3))=-4-2sqrt(6)

so you'd have to multiply again, this time by -4+2sqrt(6)

in general the "conjugate" of a+bX+cY+.... will always be a-bX-cY-.... (similar to how complex numbers work)

where a,b,c,... are from the base field and X,Y,... are the algebraic numbers you extend the field by.

Answer 4

1/(1+sqrt2+sqrt3))×

(1+sqrt2-sqrt3)/(1+sqrt2-sqrt3)=

(1+sqrt2-sqrt3)/(1+2sqrt2+2-3)=

(1+sqrt2-sqrt3)/(2sqrt2)

Multiply by sqrt2/sqrt2:

**(sqrt2+2-sqrt6)/4**

Answer 5

OP, if you get the decimal approximation of the original expression on your calculator, you can do the same for your answer and compare them.