Question

how do you simplify the expression 3 root 7 over -1- root 27? please explain with details. does multiplying by conjugates work in this problem where the radical sign is also in the numerator?

Answer 1

as you already know we'll be using the conjugate of -1-√(27), which is just -1 + √(27), so we can rationalize the denominator.

`\bf \cfrac{3√(7)}{-1-√(27)}\cdot \cfrac{-1+√(27)}{-1+√(27)}\implies \cfrac{3√(7)(-1+√(27))}{\stackrel{\textit{difference of squares}}{(-1)^2-(√(27))^2}}\implies \cfrac{-3√(7)+3√(189)}{1-27} \\\\\\ \begin{cases} 189=3\cdot 3\cdot 3\cdot 7\\ \qquad 3^2\cdot 21 \end{cases}\implies \cfrac{-3√(7)+3√(3^2\cdot 21)}{-26} \\\\\\ \cfrac{-3√(7)+9√(21)}{-26}\implies \cfrac{3√(7)-9√(21)}{26}`