Question

Which of the following is equivalent to the fraction below, when the denominator has been rationalized and x26? 12 √x - √x - 6 6 O A. 2(18+ -6) O 2(VR- O B. 2/FX-√x-6) O () C. 2(VI+Vx+6 N O D. 2(56 - X+6) +6

Answer 1

`=2(\sqrt[]{x}+\sqrt[]{x-6}_{})`

Explanation:

To rationalize the term we multiply and divide the given expression by the reciprocal:

`\frac{12}{\sqrt[]{x}-\sqrt[]{x-6}}\cdot\frac{\sqrt[]{x}+\sqrt[]{x-6}}{\sqrt[]{x}+\sqrt[]{x-6}}`

Solving:

`\begin{gathered} \frac{12}{\sqrt[]{x}-\sqrt[]{x-6}}\cdot\frac{\sqrt[]{x}+\sqrt[]{x-6}}{\sqrt[]{x}+\sqrt[]{x-6}} \\ =\frac{12(\sqrt[]{x}+\sqrt[]{x-6})}{(\sqrt[]{x}-\sqrt[]{x-6})(\sqrt[]{x}+\sqrt[]{x-6})} \\ =\frac{12(\sqrt[]{x}+\sqrt[]{x-6})}{(\sqrt[]{x})^2-(\sqrt[]{x-6})^2} \\ =\frac{12(\sqrt[]{x}+\sqrt[]{x-6})}{x-x+6} \\ =2(\sqrt[]{x}+\sqrt[]{x-6}_{}) \end{gathered}`