Question

How do you simplify:


`\sqrt{(125 ^(2) } ) ^{ - (1)/(3) }`
​

Answer 1

`\huge\boxed{\sqrt{(125^2)^{-(1)/(3)}}=(1)/(5)}`

Explanation:

`\sqrt{(125^2)^{-(1)/(3)}}\qquad\text{use}\ (a^n)^m=(a^m)^n\\\\=\sqrt{\left(125^{-(1)/(3)\right)^2}\qquad\text{use}\ √(a^2)=a\ \text{for}\ a\geq0\\\\=125^{-(1)/(3)}\qquad\text{use}\ a^(-n)=(1)/(a^n)\\\\=(1)/(125^(1)/(3))\qquad\text{use}\ a^(1)/(n)=\sqrt[n]{a}\\\\=\frac{1}{\sqrt[3]{125}}=(1)/(5)\qquad\text{because}\ 5^3=125`