Rationalize the denominator and simplify.
-$\frac { \sqrt { 2 } - 7 } { \sqrt { 7 } - 10 }$
A) $\frac { \sqrt { 14 } + 70 - 7 \sqrt { 7 } - 10 \sqrt { 2 } } { - 93 }$
B) $\frac { \sqrt { 14 } - 70 - 7 \sqrt { 7 } + 10 \sqrt { 2 } } { 107 }$
C) $\frac { \sqrt { 14 } - 70 - 7 \sqrt { 7 } + 10 \sqrt { 2 } } { - 93 }$
D) $\frac { \sqrt { 14 } + 70 - 7 \sqrt { 7 } - 10 \sqrt { 2 } } { 107 }$

Question

Quizplus · Accepted Answer

To rationalize the denominator, we need to multiply both the numerator and denominator of the fraction by the conjugate of the denominator, which is $\sqrt{7}+10$.
\begin{align*}
\frac{\sqrt{2}-7}{\sqrt{7}-10}&=\frac{(\sqrt{2}-7)(\sqrt{7}+10)}{(\sqrt{7}-10)(\sqrt{7}+10)}\
&=\frac{\sqrt{14}+10\sqrt{2}-7\sqrt{7}-70}{-93}\
&=\frac{\sqrt{14}-7\sqrt{7}+10\sqrt{2}-70}{-93}\
&=\frac{\sqrt{14}-70-7\sqrt{7}+10\sqrt{2}}{-93}
\end{align*}
Thus, the correct choice is C.

Rationalize the Denominator and Simplify 2−77−10\frac { \sqrt { 2 } - 7 } { \sqrt { 7 } - 10 }7​−102​−7​

Rationalize the Denominator and Simplify $\frac { \sqrt { 2 } - 7 } { \sqrt { 7 } - 10 }$