(a) Express (1+4*sqrt(7))/(5+2*sqrt(7)) in the form a+b*sqrt(7), where a and b are integers. (b) Then solve the equation x*(9*sqrt(5)-2*sqrt(45))=sqrt(80).

(a)   We can ‘get rid of’ a square root in the denominator simply by multiplying by 1 (value of the fraction stays unchanged) in a suitable form. We will take advantage of this formula: (a+b)(a-b)=a^2-b^2. So (1+4sqrt(7))/(5+2sqrt(7))= (1+4sqrt(7))/(5+2sqrt(7))1=(1+4sqrt(7))/(5+2sqrt(7))(5-2sqrt(7))/( 5-2sqrt(7))=(5+20sqrt(7)-2sqrt(7)-56)/(25+10sqrt(7)-10sqrt(7)-28)=(-51+18sqrt(7))/(-3)=17-6sqrt(7). The question asks for integers a and b, so the final step is: a=17, b=-6.(b)   Now we know how to eliminate square roots from the denominator, we can use this to solve equations like this one. First we will express x:x=sqrt(80)/ (9sqrt(5)-2sqrt(45))= sqrt(80)/ (9sqrt(5)-2sqrt(45))((9sqrt(5)+2sqrt(45))/ (9sqrt(5)+2sqrt(45)))=(9sqrt(805)+2sqrt(4580))/(815-445)=(180+120)/225=4/3.