simplify the radical expression: \n$\\sqrt{27m^{7}p^{4}}$

Explanation:

Step1: Factor the radicand

$$ \begin{align*} \sqrt{27m^{7}p^{4}}&=\sqrt{9\times3\times m^{6}\times m\times p^{4}}\ \end{align*} $$

Step2: Use the property $\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}$

$$ \begin{align*} \sqrt{9\times3\times m^{6}\times m\times p^{4}}&=\sqrt{9}\cdot\sqrt{m^{6}}\cdot\sqrt{p^{4}}\cdot\sqrt{3m}\ \end{align*} $$

Step3: Simplify each square - root

Since $\sqrt{9} = 3$, $\sqrt{m^{6}}=m^{3}$ ($(m^{3})^{2}=m^{6}$), $\sqrt{p^{4}}=p^{2}$ ($(p^{2})^{2}=p^{4}$) $$ \begin{align*} \sqrt{9}\cdot\sqrt{m^{6}}\cdot\sqrt{p^{4}}\cdot\sqrt{3m}&=3\cdot m^{3}\cdot p^{2}\cdot\sqrt{3m}\ &=3m^{3}p^{2}\sqrt{3m} \end{align*} $$

Answer:

$3m^{3}p^{2}\sqrt{3m}$ (the fourth option)