Question

Which choice is equivalent to the product below when x> 0? 3x xli 3x ОАА O A. ОВ. Ос. C. Ž O D. D. x 2.

Answer 1

We can use the following properties of radicals:

`\begin{gathered} \sqrt[n]{ab}=\sqrt[n]{a}\cdot\sqrt[n]{b}\Rightarrow\text{ Product property} \\ \sqrt[n]{(a)/(b)}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}\Rightarrow\text{ Quotient property} \end{gathered}`

Then, we have:

`\begin{gathered} \text{ Apply the product property} \\ \sqrt[]{(3x)/(2)}\cdot\sqrt[]{(x)/(6)}=\sqrt[]{(3x\cdot x)/(2\cdot6)} \\ \sqrt[]{(3x)/(2)}\cdot\sqrt[]{(x)/(6)}=\sqrt[]{(3x^2)/(12)} \\ \text{ Simplify the expression inside the radical} \\ \sqrt[]{(3x)/(2)}\cdot\sqrt[]{(x)/(6)}=\sqrt[]{(3x^2)/(3\cdot4)} \\ \sqrt[]{(3x)/(2)}\cdot\sqrt[]{(x)/(6)}=\sqrt[]{(x^2)/(4)} \\ \text{ Apply the quotient property} \\ \sqrt[]{(3x)/(2)}\cdot\sqrt[]{(x)/(6)}=\frac{\sqrt[]{x^2}}{\sqrt[]{4}} \\ \sqrt[]{(3x)/(2)}\cdot\sqrt[]{(x)/(6)}=(x)/(2) \end{gathered}`

Therefore, the choice that is equivalent to the given product when x > 0 is:

`(x)/(2)`