Not taking any shortcuts so you can see what is happening.
Given: #3xsqrt(8x) - 4 sqrt(x^3)+3xsqrt(72x)color(white)(..) # ...... (1)
We may choose two approaches.
#color(blue)("Approach 1")#
Just do a strait substitution and calculate. This will most likely yield an imprecise decimal solution. Note they say "what is the value". Not "what is the approximate value".
#color(blue)("Approach 2")#
Simplify and stick with surds so that any answer is guaranteed to be precise.
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Adopting approach 2")#
Looking for factors that can be used to simplify:
Notice that #9xx8 =72# so we have a similarity within the expression of #8x#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(brown)("Consider: "-4sqrt(x^3) )# If you take the 4 inside the root you have to square it.
Write as : # -sqrt(16x xx x^2)=-sqrt(8x xx 2x^2)#
#-=-(xsqrt(2) xx sqrt(8x))#........................(2)
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(brown)("Consider: "+3xsqrt(72x) )#
Write as: #+3x sqrt(9 xx8x) -= +9xsqrt(8x)#......(3)
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Substitute (2 ) and (3 ) into (1 ) giving:")#
#3xsqrt(8x) -(xsqrt(2) xx sqrt(8x))+9xsqrt(8x)#........(4)
#color(brown)("Factoring out the "sqrt(8x))#
#sqrt(8x)color(white)(.)( 3x - sqrt(2 )color(white)(.)x+9x)#
#sqrt(8x)color(white)(.)( 12x - sqrt(2 )color(white)(.)x)#
#color(brown)("Factoring out the "x " from the btackets")#
#xsqrt(8x)(12-sqrt(2))#.........................(5)
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Substitute "x=2 " in (5 )")#
#(2)(sqrt(16))(12-sqrt(2))#
#8(12-sqrt(2))#
#=>96 - 8 sqrt(2)#
#color(brown)("As an approximate decimal value this is "84.69" to 2 decimal places.")#
#color(green)("To an EXACT value this is "=>96 - 8 sqrt(2))#
