Question

What is equivalent to (81m^6)^1/2

Answer 1

`\large\boxed{(81m^6)^(1)/(2)=9m^3}`

Explanation:

`81=9^2\\\\m^6=m^(3\cdot2)=(m^3)^2\qquad\text{used}\ (a^n)^m=a^(nm)\\\\\left(81m^6\right)^(1)/(2)=\bigg(9^2\left(m^3\right)^2\bigg)^(1)/(2)\qquad\text{use}\ (ab)^n=a^nb^n\\\\=\left(9^2\right)^(1)/(2)\bigg(\left(m^3\right)^2\bigg)^(1)/(2)\qquad\text{use}\ (a^n)^m=a^(nm)\\\\=9^{2\cdot(1)/(2)}\left(m^3\right)^{2\cdot(1)/(2)}=9^1(m^3)^1=9m^3`

Other method:

`Use\ a^(1)/(n)=\sqrt[n]{a}\to a^(1)/(2)=\sqrt[2]{a}=√(a)\\\\\left(81m^6\right)^(1)/(2)=√(81m^6)\qquad\text{use}\ √(ab)=√(a)\cdot√(b)\ \text{and}\ (a^n)^m=a^(nm)\\\\=√(81)\cdot\sqrt{m^(3\cdot2)}=9√((m^3)^2)\qquad\text{use}\ \sqrt[n]{a^n}=a\to√(a^2)=a\\\\=9m^3`