$$
\begin{aligned}
& \sqrt{x} = y \ \Longleftrightarrow \ x = y^2 \\ \\
& \sqrt[n]{x} = y \ \Longleftrightarrow \ x = y^n
\end{aligned}
$$
$$
\begin{aligned}
& \sqrt{x} = x^{\frac{1}{2}} \\ \\
& \sqrt[n]{x^m} = x^{\frac{m}{n}}
\end{aligned}
$$
$$
\begin{aligned}
& \sqrt[p \cdot n]{x^{p \cdot m}} = \sqrt[n]{x^m} = x^{\frac{m}{n}} \\ \\
& \left(\sqrt[n]{x}\right)^m = \sqrt[n]{x^m} \\ \\
& \sqrt[n]{\sqrt[m]{x}} = \sqrt[m]{\sqrt[n]{x}} = \sqrt[m \cdot n]{x}
\end{aligned}
$$
$$
\begin{aligned}
& \sqrt[n]{x \cdot y} = \sqrt[n]{x} \cdot \sqrt[n]{y} \\ \\
& \sqrt[n]{\frac{x}{y}} = \frac{\sqrt[n]{x}}{\sqrt[n]{y}}
\end{aligned}
$$
$$
\begin{aligned}
& x^n \cdot x^p = x^{n + p} \\ \\
& \frac{x^n}{x^p} = x^{n - p} \\ \\
& \left(x^n\right)^p = x^{n \cdot p} \\ \\
& x^{-n} = \frac{1}{x^n}
\end{aligned}
$$
\begin{aligned}
& x^n \cdot y^n = \left(x \cdot y\right)^n \\ \\
& \frac{x^n}{y^n} = \left(\frac{x}{y}\right)^n
\end{aligned}
$$
\begin{aligned}
& x^0 = 1 \\ \\
& x^1 = x
\end{aligned}
$$
$$
\begin{aligned}
& 0^n = 0 \\ \\
& 1^n = 1
\end{aligned}
$$
