$\sqrt{ 125 \times \left( -1 \right) }$
The root of a product is equal to the product of the roots of each factor$\sqrt{ 125 }\sqrt{ -1 }$
Write the number in exponential form with the base of $5$$\sqrt{ {5}^{3} }\sqrt{ -1 }$
Use $\sqrt[2]{-1}=i$ to simplify the expression$\sqrt{ {5}^{3} }i$
Rewrite the exponent as a sum where one of the addends is a multiple of the index$\sqrt{ {5}^{2+1} }i$
Use ${a}^{m+n}={a}^{m} \times {a}^{n}$ to expand the expression$\sqrt{ {5}^{2} \times {5}^{1} }i$
Any expression raised to the power of $1$ equals itself$\sqrt{ {5}^{2} \times 5 }i$
The root of a product is equal to the product of the roots of each factor$\sqrt{ {5}^{2} }\sqrt{ 5 }i$
Reduce the index of the radical and exponent with $2$$\begin{align*}&5\sqrt{ 5 }i \\&\approx11.18034i\end{align*}$