Answer:
[tex]\large\boxed{(81m^6)^\frac{1}{2}=9m^3}[/tex]
Step-by-step explanation:
[tex]81=9^2\\\\m^6=m^{3\cdot2}=(m^3)^2\qquad\text{used}\ (a^n)^m=a^{nm}\\\\\left(81m^6\right)^\frac{1}{2}=\bigg(9^2\left(m^3\right)^2\bigg)^\frac{1}{2}\qquad\text{use}\ (ab)^n=a^nb^n\\\\=\left(9^2\right)^\frac{1}{2}\bigg(\left(m^3\right)^2\bigg)^\frac{1}{2}\qquad\text{use}\ (a^n)^m=a^{nm}\\\\=9^{2\cdot\frac{1}{2}}\left(m^3\right)^{2\cdot\frac{1}{2}}=9^1(m^3)^1=9m^3[/tex]
Other method:
[tex]Use\ a^\frac{1}{n}=\sqrt[n]{a}\to a^\frac{1}{2}=\sqrt[2]{a}=\sqrt{a}\\\\\left(81m^6\right)^\frac{1}{2}=\sqrt{81m^6}\qquad\text{use}\ \sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\ \text{and}\ (a^n)^m=a^{nm}\\\\=\sqrt{81}\cdot\sqrt{m^{3\cdot2}}=9\sqrt{(m^3)^2}\qquad\text{use}\ \sqrt[n]{a^n}=a\to\sqrt{a^2}=a\\\\=9m^3[/tex]
