Question

Find the value of a and b

Answer 1

`\large\boxed{a=(58)/(25),\ b=0}`

Explanation:

`(3\sqrt3+\sqrt2)/(3\sqrt3-\sqrt2)+(3\sqrt3-\sqrt2)/(3\sqrt3+\sqrt2)\qquad\text{use}\ (a-b)(a+b)=a^2-b^2\\\\=((3\sqrt3+\sqrt2)(3\sqrt3+\sqrt2))/((3\sqrt3-\sqrt2)(3\sqrt3+\sqrt2))+((3\sqrt3-\sqrt2)(3\sqrt3-\sqrt2))/((3\sqrt3+\sqrt2)(3\sqrt3-\sqrt2))\\\\=((3\sqrt3+\sqrt2)^2)/((3\sqrt3)^2-(\sqrt2)^2)+((3\sqrt3-\sqrt2)^2)/((3\sqrt3)^2-(\sqrt2)^2)\qquad\text{use}\ (a\pm b)^2=a^2\pm2ab+b^2`

`=((3\sqrt3)^2+2(3\sqrt3)(\sqrt2)+(\sqrt2)^2)/(3^2(\sqrt3)^2-2)+((3\sqrt3)^2-2(3\sqrt3)(\sqrt2)+(\sqrt2)^2)/(3^2(\sqrt3)^2-2)\\\\=(3^2(\sqrt3)^2+6\sqrt6+2)/(9\cdot3-2)+(3^2(\sqrt3)^2-6\sqrt6+2)/(9\cdot3-2)\\\\=(9\cdot3+6\sqrt6+2)/(27-2)+(9\cdot3-6\sqrt6+2)/(27-2)\\\\=(27+6\sqrt6+2)/(25)+(27-6\sqrt6+2)/(25)\\\\=(29+6\sqrt6)/(25)+(29-6\sqrt6)/(25)\\\\=(29+6\sqrt6+29-6\sqrt6)/(25)\\\\=(58)/(25)`