Question 139033
I presume you mean {{{(3sqrt(3))/(3sqrt(7))}}}


That 'division symbol only it has a tail' is the square root symbol.  It means 'find me a number when multiplied by itself equals the number under the symbol'  Unless the number under the symbol is a perfect square, like 4, 9, 16, 25..., then the square root is an irrational number.  A rational number is one that can be expressed as the quotient of two integers, {{{p/q}}} where p and q are integers.  An irrational number cannot be expressed this way.


So rationalizing a denominator means 'get that nasty radical (another term for the square root symbol) the heck out of my denominator'


In this problem, we need to multiply the denominator by {{{sqrt(7)}}}, so the denominator will become {{{3*sqrt(7)*sqrt(7)=3*7=21}}}.  BUT, we have to do so in such a way that we don't change the value of the overall fraction.  Since we know that anything multiplied by 1 is itself ({{{a*1=a}}} no matter what a is), we can multiply the fraction by 1 in the form of {{{sqrt(7)/sqrt(7)}}}. ({{{a/a=1}}} no matter what a is).


But first a rule about radicals:  No matter what a and b are (as long as they are positive) we can say that {{{sqrt(a)*sqrt(b)=sqrt(ab)}}}.


Now:


{{{((3sqrt(3))/(3sqrt(7)))((sqrt(7))/(sqrt(7)))=(3*sqrt(3)*sqrt(7))/(3*sqrt(7)*sqrt(7))=3*sqrt(21)/21}}}.


So we have a new fraction that is equivalent to the original, but has a rational number (21) in the denominator.  Done.