Evaluate: sqrt(-125)

Expression: $\sqrt{ -125 }$

Write the number as a product with the factor $-1$

$\sqrt{ 125 \times \left( -1 \right) }$

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 $2$

$\begin{align*}&5\sqrt{ 5 }i \\&\approx11.18034i\end{align*}$