Question #07db7

We can start by simplifying our surds.

#3/(sqrt48-sqrt75)#

#3/(sqrt4sqrt12-sqrt25sqrt3)#

#3/(2sqrt12-5sqrt3)#

#3/(2sqrt4sqrt3-5sqrt3)#

#3/(2xx2sqrt3-5sqrt3)#

Simplify the bottom by collecting like terms.

#3/(4sqrt3-5sqrt3)#

#3/(-sqrt3)#

Now we need to rationalise the denominator. By multiplying the top and bottom by the surd at the bottom.

#3/(-sqrt3)xxsqrt3/sqrt3#

#(3sqrt3)/(-sqrt3sqrt3)#

#(3sqrt3)/(-sqrt9)#

#(3sqrt3)/(-3)#

And cancel the three's.

#(cancel3sqrt3)/(-cancel3)#

#-sqrt3#

The trick with these is to look for squared values within the value you are taking the square root of. Then use those to try and determine common values that you can manipulate.

Consider 48 #->3xx16->3xx4^2#
Consider 75 #->3xx25->3xx5^2#
Write as:

#3/(sqrt48 - sqrt75) color(white)("d")->color(white)("d") 3/(sqrt(3xx4^2)-sqrt(3xx5^2))#

#color(white)("dddddddddd")->color(white)("ddddd")3/(4sqrt(3)-5sqrt(3)) #

#color(white)("dddddddddd")->color(white)("ddddddd")3/(-sqrt3)#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Mathematicians do not like roots in the denominator if they can help it. So lets get rid of it.

By the way: #3/(-sqrt3)# is the same as #-3/sqrt3#

#color(green)(-3/sqrt3color(red)(xx1) color(white)("d")=color(white)("d")-3/sqrt3color(red)(xxsqrt3/sqrt3#

#color(white)("ddddddddd")=color(white)("ddd")-(cancel(3)^1sqrt3)/(cancel(3)^1)#

#color(white)("ddddddddd")=color(white)("ddd")-sqrt3#

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Check")#

#3/(sqrt48-sqrt75) ~~-1.73205.......... #

#color(white)("d")-sqrt3 color(white)("ddddd")~~ -1.73205 .........color(red)(larr" Thus solution ok.")#