Rationalize the denominator and simplify. The variable represents a positive real number. $\frac { \sqrt { x } } { \sqrt { x } - 10 }$
A) $\frac { 10 \sqrt { x } } { x - 100 }$
B) $\frac { x + 10 \sqrt { x } } { x - 10 }$
C) $\sqrt { x }$
D) $\frac { 11 x } { x - 100 }$
E) $\frac { x + 10 \sqrt { x } } { x - 100 }$

Question

Quizplus · Accepted Answer

To rationalize the denominator of $\frac { \sqrt { x } } { \sqrt { x } - 10 }$, multiply both the numerator and the denominator by the conjugate of the denominator, which is $\sqrt { x } + 10$. This gives:$$\frac { \sqrt { x } } { \sqrt { x } - 10 } \cdot \frac { \sqrt { x } + 10 } { \sqrt { x } + 10 } = \frac { \sqrt { x }(\sqrt { x } + 10) } { (\sqrt { x } - 10)(\sqrt { x } + 10) }$$Simplifying the numerator and the denominator separately:Numerator: $\sqrt { x } \cdot \sqrt { x } + \sqrt { x } \cdot 10 = x + 10\sqrt { x }$Denominator: $(\sqrt { x })^2 - (10)^2 = x - 100$Thus, the simplified expression is $\frac { x + 10 \sqrt { x } } { x - 100 }$, which matches choice E.

Rationalize the Denominator and Simplify xx−10\frac { \sqrt { x } } { \sqrt { x } - 10 }x​−10x​​

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