Rationalize the denominator and simplify.

Question

Quizplus · Accepted Answer

To rationalize the denominator, we need to multiply both the numerator and denominator by the conjugate of the denominator, which is $\sqrt{11ebb939_4a44_815e_9e6c_9f7bd1a5befe_TB8446_11} + \sqrt{11ebb939_4a44_815e_9e6c_9f7bd1a5befe_TB8446_11}$. This gives us:
$$\frac{1}{\sqrt{11ebb939_4a44_815e_9e6c_9f7bd1a5befe_TB8446_11} - \sqrt{11ebb939_4a44_815e_9e6c_9f7bd1a5befe_TB8446_11}} \cdot \frac{\sqrt{11ebb939_4a44_815e_9e6c_9f7bd1a5befe_TB8446_11} + \sqrt{11ebb939_4a44_815e_9e6c_9f7bd1a5befe_TB8446_11}}{\sqrt{11ebb939_4a44_815e_9e6c_9f7bd1a5befe_TB8446_11} + \sqrt{11ebb939_4a44_815e_9e6c_9f7bd1a5befe_TB8446_11}}$$
Simplifying the denominator gives us:
$$(\sqrt{11ebb939_4a44_815e_9e6c_9f7bd1a5befe_TB8446_11})^2 - (\sqrt{11ebb939_4a44_815e_9e6c_9f7bd1a5befe_TB8446_11})^2 + 2\sqrt{11ebb939_4a44_815e_9e6c_9f7bd1a5befe_TB8446_11}\sqrt{11ebb939_4a44_815e_9e6c_9f7bd1a5befe_TB8446_11}$$
which simplifies to:
$$2\sqrt{11ebb939_4a44_815e_9e6c_9f7bd1a5befe_TB8446_11}$$
Therefore, the simplified form is:
$$\frac{\sqrt{11ebb939_4a44_815e_9e6c_9f7bd1a5befe_TB8446_11} + \sqrt{11ebb939_4a44_815e_9e6c_9f7bd1a5befe_TB8446_11}}{2\sqrt{11ebb939_4a44_815e_9e6c_9f7bd1a5befe_TB8446_11}}$$
which further simplifies to:
$$\frac{1}{2}$$
Therefore, the correct answer is C. To rationalize the denominator, we need to multiply both the numerator and denominator by the conjugate of the denominator, which is $\sqrt{11ebb939_4a44_815e_9e6c_9f7bd1a5befe_TB8446_11} + \sqrt{11ebb939_4a44_815e_9e6c_9f7bd1a5befe_TB8446_11}$.
This gives us:
$$\frac{1}{\sqrt{11ebb939_4a44_815e_9e6c_9f7bd1a5befe_TB8446_11} - \sqrt{11ebb939_4a44_815e_9e6c_9f7bd1a5befe_TB8446_11}}} 	imes \frac{\sqrt{11ebb939_4a44_815e_9e6c_9f7bd1a5befe_TB8446_11} + \sqrt{11ebb939_4a44_815e_9e6c_9f7bd1a5befe_TB8446_11}}{\sqrt{11ebb939_4a44_815e_9e6c_9f7bd1a5befe_TB8446_11} + \sqrt{11ebb939_4a44_815e_9e6c_9f7bd1a5befe_TB8446_11}}}$$
Simplifying the numerator gives us $1$, and simplifying the denominator gives us $\sqrt{11ebb939_4a44_a871_9e6c_650a129b36f1_TB8446_11}$, which is the answer choice C.

Rationalize the Denominator and Simplify