square root (should be red): $\sqrt{2}$

long square root: $\sqrt{x+1}$

long square root with implied row: $\sqrt{x^2+1}$

root of a fraction: $\sqrt{\frac{x+1}{x+2}}$

cube root (should be blue): $\sqrt[3]{2}$

long index: $\sqrt[x+y+z]{2}$

long index w/ complex base: $\sqrt[x+y+z]{\frac{x+1}{x+2}}$

high index: $\sqrt[\frac{\frac{x}{y}}{z}]{2}$

Imbricated square roots: $\sqrt{1+\sqrt{2+\sqrt{3+\sqrt{4+\sqrt{5+\sqrt{6+\sqrt{7+\sqrt{A}}}}}}}}$

Imbricated roots: $\sqrt[3]{1+\sqrt[4]{2+\sqrt[5]{3+\sqrt[6]{4+\sqrt[7]{5+\sqrt[8]{6+\sqrt[9]{7+\sqrt[\frac{x+y}{z}]{A}}}}}}}}$

Don't wrap this wide line with an implicit mrow:
$\sqrt{xy}$