ROOTS

ROOTS, real and imaginary. The odd roots, as the 3d, 5th, 7th, &c. of all real quantities, whether positive or negative, are real, and are respectively positive or negative. So the cube root of $a^3$ is $a$, and of $-a^3$ is $-a$. But the even roots, as the 2d, 4th, 6th, &c. are only real when the quantity is positive, being imaginary or impossible when the quantity is negative. So the square root of $a^2$ is $a$, which is real; but the square root of $-a^2$, that is, $\backslash sqrt\{-a^2\}$, is imaginary or impossible, because there is no quantity, neither $+a$ nor $-a$, which by squaring will make the given negative square $-a^2$.