Question

This is about SURDS I need a detailed answer before tomorrow 8pm
see to the attached file​

Answer 1

let's recall that anything multiplied by "1" is simply itself as a product, and that "1" can be a fraction with the same over the same, namely

`\cfrac{25}{25}\implies \cfrac{999999}{999999}\implies \cfrac{cheese}{cheese}\implies \cfrac{spaghetti}{spaghetti}\implies \text{\LARGE 1}`

so let's multiply the rational by the conjugate of the denominator in order to rationalize the denominator.

`\cfrac{3+√(2)}{5+√(8)}\cdot \text{\LARGE 1}\implies \cfrac{3+√(2)}{5+√(8)}\cdot \cfrac{5-√(8)}{5-√(8)}\implies \cfrac{(3+√(2))(5-√(8))}{\underset{ \textit{difference of squares} }{(5+√(8))(5-√(8))}} \\\\\\ \cfrac{(3+√(2))(5-√(8))}{(5)^2~~ - ~~(√(8))^2}\implies \cfrac{15-3√(8)+5√(2)-√(2)\cdot √(8)}{25~~ - ~~8}`

`\cfrac{15-3√(8)+5√(2)-√(16)}{17}\implies \cfrac{15-3√(2^2\cdot 2)+5√(2)-√(16)}{17} \\\\\\ \cfrac{15-6√(2)+5√(2)-4}{17}\implies \cfrac{15-4-6√(2)+5√(2)}{17}\implies \cfrac{11-√(2)}{17}`