The expression to simplify is:
![9\sqrt[]{2}(4\sqrt[]{6})](https://img.qammunity.org/2023/formulas/mathematics/college/ryhwse0mnwemcalfjgor5z4027m00rmf87.png)
When we are multiplying two racial expressions, we multiply the constants together and the square roots together. So, the next step is:
![\begin{gathered} 9\sqrt[]{2}(4\sqrt[]{6}) \\ =(9*4)(\sqrt[]{2}*\sqrt[]{6}) \\ =36(\sqrt[]{2}*\sqrt[]{6}) \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/college/fu48zm302ormnhz9ujjdamcro5k4bwz0lp.png)
Now, we an use the property
![\sqrt[]{a}*\sqrt[]{b}=\sqrt[]{a* b}](https://img.qammunity.org/2023/formulas/mathematics/college/chqy4dz8n8x5nioc2edok2vfwyna78i85y.png)
to simplify it further:
![\begin{gathered} 36(\sqrt[]{2}*\sqrt[]{6}) \\ =36(\sqrt[]{2*6}) \\ =36\sqrt[]{12} \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/college/exj7h1bfbz1kuiib9ooxafm2dx83kq41dd.png)
We can break apart the square root using the property:
![\sqrt[]{ab}=\sqrt[]{a}\sqrt[]{b}](https://img.qammunity.org/2023/formulas/mathematics/college/36x8mrtqxls717blnt9mq87duprn6qlre4.png)
So, we have:
![\begin{gathered} 36\sqrt[]{12} \\ =36\sqrt[]{2}\sqrt[]{2}\sqrt[]{3} \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/college/299xd6nrr0nnjv8kbuvjs7c4120zf4xdvv.png)
For the final simplification, we use the property,
![\sqrt[]{a}\sqrt[]{a}=a](https://img.qammunity.org/2023/formulas/mathematics/college/9w25u8pls7bk3geaba1ya2wuk0wz3qk7nq.png)
The final answer is:
![\begin{gathered} 36\sqrt[]{2}\sqrt[]{2}\sqrt[]{3} \\ =36(2)\sqrt[]{3} \\ =72\sqrt[]{3} \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/college/rkkzfruwlct0mh1ykgnbtwbqem3fomtsmh.png)
Answer
![72\sqrt[]{3}](https://img.qammunity.org/2023/formulas/mathematics/college/76d8fkufw6c9h90htqxqweyq55ywxmmzmn.png)