If both `a\ a n d\ b` are rational numbers, find the values of `a\ a n d\ b` in each of the following equalities: `(5+sqrt(3))/(7-4sqrt(3))=47\ a+\ sqrt(3)\ b`

Explanation

We have to rationalize the denominator to know the value of a and b.
`(5+√3) /(7-4√3)`
=`[(5+√3)*(7+4√3)] /[(7-4√3)*(7+4√3)]`
=`(5+√3)(7+4√3)/[(7-4√3)(7+4√3)]`
=`[35 +20√3 +7√3 +4√3 (√3)]/[(7)²-(4√3)²]`
=`(35 +27√3+4(3))/[49-16(3)]`
=`(35+27√3+12)/[49-48]`
=`47+27√3/1`
=`47+27√3`=`47a+√3b`
We can write it as
`47(1)+√3(27)`
So,`a=1` and `b=27`.