Question

Rationalize the denominator of the fraction below. What is the new denominator? 5 3 + V6 O A. 3 B. -3 O C.-27 O D. 15

Answer 1

We multiply the numerator and denominator by the conjugate of the denominator to rationalize it. The conjugate of the denominator is the same expression but with the opposite sign.

`\begin{gathered} \text{ Denominator }=3+\sqrt[]{6} \\ \text{ Conjugate of the denominator }=3-\sqrt[]{6} \end{gathered}`

Then, we have:

`\begin{gathered} \frac{5}{3+\sqrt[]{6}}=\frac{5}{3+\sqrt[]{6}}\cdot\frac{3-\sqrt[]{6}}{3-\sqrt[]{6}} \\ \frac{5}{3+\sqrt[]{6}}=\frac{5(3-\sqrt[]{6})}{(3+\sqrt[]{6})(3-\sqrt[]{6})} \end{gathered}`

Now, we apply the difference of squares formula.

`(a-b)(a+b)=a^2-b^2`
`\begin{gathered} \frac{5}{3+\sqrt[]{6}}=\frac{5\cdot3-5\cdot\sqrt[]{6}}{3^2-(\sqrt[]{6})^2} \\ \frac{5}{3+\sqrt[]{6}}=\frac{15-5\sqrt[]{6}}{9-6} \\ \frac{5}{3+\sqrt[]{6}}=\frac{15-5\sqrt[]{6}}{3} \end{gathered}`

Therefore, after rationalizing the denominator of the given fraction, the new denominator is 3.