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.

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