How do you evaluate #-\sqrt { 18} - 3\sqrt { 6} + 3\sqrt { 24}#?

We can use this rule for radicals to rewrite the term on the left of the expression:

#sqrt(a * b) = sqrt(a) * sqrt(b)#

#-sqrt(18) - 3sqrt(6) + 3sqrt(24) => -sqrt(9 * 2) - 3sqrt(6) + 3sqrt(24) =>#

#-(sqrt(9) * sqrt(2)) - 3sqrt(6) + 3sqrt(24) =>#

#-3sqrt(2) - 3sqrt(6) + 3sqrt(24)#

We can next factor out a #3# from each term in the expression:

#(3 * -sqrt(2)) + (3 * -sqrt(6)) + (3 * sqrt(24)) =>#

#3(-sqrt(2) - sqrt(6) + sqrt(24))#

Now, rewrite #sqrt(24)# to give:

#3(-sqrt(2) - sqrt(6) + sqrt(24)) => 3(-sqrt(2) - sqrt(6) + sqrt(4 * 6)) => #

#3(-sqrt(2) - sqrt(6) + (sqrt(4) * sqrt(6))) => 3(-sqrt(2) - sqrt(6) + 2sqrt(6)) => 3(-sqrt(2) + (-1sqrt(6) + 2sqrt(6)) => 3(-sqrt(2) + sqrt(6))#

Now, rewrite #sqrt(6)# to give:

#3(-sqrt(2) + sqrt(6)) => 3(-sqrt(2) + sqrt(2 * 3)) =>#

#3(-sqrt(2) + (sqrt(2) * sqrt(3))) => 3(-1sqrt(2) + (sqrt(2) * sqrt(3))) =>#

#3((-1+ sqrt(3))sqrt(2)) =>#

#3sqrt(2)(sqrt(3) - 1)#

Always worth looking for common factors and seeing what happens. In roots it is also always looking for squared factors you can 'take outside' the roots.

#18-> 2xx 9 -> 2xx3^2#
#6-> 2 xx 3#
#24->2xx12->2xx3xx4->2xx3xx2^2#

Note that #sqrt(2xx3)# is the same as #sqrt(2)xxsqrt(3)#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(green)(-sqrt18)color(red)(-3sqrt6)color(blue)(+3sqrt24)#

#color(green)(-sqrt(2xx3^2))color(red)(-3sqrt(2xx3)color(blue)(+3sqrt(3xx2xx2^2)#

#color(green)([-3sqrt(2)])color(red)(-[3sqrt(2)xxsqrt(3)]color(blue)(+[6sqrt(3)xxsqrt(2)])#

But #6sqrt3 xx sqrt2# is the same as #3sqrt2xx2sqrt3#
Factoring out #-3sqrt2# gives:

#-3sqrt(2) (1+sqrt3-2sqrt(3))#

#-3sqrt(2)(1-sqrt(3))#

#3sqrt2(sqrt(3)-1)#

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
this is approximately #3.1058285...#

#-sqrt18-3sqrt6+3sqrt24~~3.1058285....#