Radical (Math)

Both the radicand R and the order of the root n have an effect on the root(s) P. For example, because a negative number multiplied by a negative number is a positive number, the even roots (n = 2, 4, 6, 8...) of a positive number are both negative and positive: √9 = ±3, &NA; = ±2.

Because the root P of &NA; must be multiplied an odd number of times to generate the radicand R, it should be clear that the odd roots (n = 3, 5, 7, 9...) of a positive number are positive, and the odd roots of a negative number are negative. For example, &NA; = 2 (2³ = 2 × 2 × 2 = 4 × 2 = 8), but &NA; = −2 (−2³ = −2 × −2 × −2 = 4 × −2 = 8).

Taking an even root of a negative number is a trickier business altogether. As discussed above, the product of an even number of negative values is a positive number. The even root of a negative number is imaginary. That is, we define the imaginary unit i = √−1 or 2 i = −1. Then √−9 = √9 × √−1 = ±3i. The imaginary unit is a very useful concept in certain types of calculus and complex analysis.