Resposta :
Supomos que o valor da expressão seja [tex]k[/tex], então:
[tex]\sqrt{\frac{a}{b}}+\sqrt{\frac{b}{a}}=k\\\\\\(\sqrt{\frac{a}{b}}+\sqrt{\frac{b}{a}})^2=k^2\\\\\\\frac{a}{b}+2\sqrt{\frac{a}{b}\cdot\frac{b}{a}}+\frac{b}{a}=k^2\\\\\\\frac{a}{b}+\frac{b}{a}+2=k^2\\\\\\k^2=\frac{a^2+b^2}{ab}+2\\\\\\k^2=\frac{(a+b)^2-2ab}{ab}+2\\\\\\k^2=\frac{4^2-2\cdot2}{2}+2[/tex]
[tex]k^2=\frac{16-4}{2}+2\\\\k^2=\frac{12}{2}+2\\\\k^2=8\\\\k=\sqrt{8}\\\\\boxed{k=2\sqrt{2}}[/tex]
[tex]\sqrt{\frac{a}{b}}+\sqrt{\frac{b}{a}}=k\\\\\\(\sqrt{\frac{a}{b}}+\sqrt{\frac{b}{a}})^2=k^2\\\\\\\frac{a}{b}+2\sqrt{\frac{a}{b}\cdot\frac{b}{a}}+\frac{b}{a}=k^2\\\\\\\frac{a}{b}+\frac{b}{a}+2=k^2\\\\\\k^2=\frac{a^2+b^2}{ab}+2\\\\\\k^2=\frac{(a+b)^2-2ab}{ab}+2\\\\\\k^2=\frac{4^2-2\cdot2}{2}+2[/tex]
[tex]k^2=\frac{16-4}{2}+2\\\\k^2=\frac{12}{2}+2\\\\k^2=8\\\\k=\sqrt{8}\\\\\boxed{k=2\sqrt{2}}[/tex]