Radical (Math)

A radical is a symbol for the indicated root of a number, for example a square root or cube root; the term is also synonymous for the root itself.

The word radical has both Latin and Greek origins. From Latin raidix, radicis means "root" and in Greek raidix is the analog word for "branch." The concept of a radical—the root of a number—can best be understood by first tackling the idea of exponentiation, or raising a number to a given power. We indicate a number raised to the nth power by writing xⁿ. This expression indicates that we are multiplying x by itself n number of times. For example, 3² = 3 × 3 = 9, and 2⁴ = 2 × 2 × 2 × 2 = 16.

Just as division is the inverse of multiplication, taking the root of a number is the inverse of raising a number to a power. For example, if we are seeking the square root of x², which equals x × x, then we are seeking the variable that, when multiplied by itself, is equal to x²—namely, x. That is to say, √9 = 3² = 3 × 3. Similarly, if we are looking for the fourth root of x ⁴, then we are looking for the variable that multiplied by itself four times equals x. For example, [fourth root of 16] x = 2⁴ = 2 × 2 × 2 × 2.

The radical &NA; is the symbol that calls for the root operation; the number or variable under the radical sign is called the radicand. It is common parlance to speak of the radicand as being "under the radical." It is also common to simply use the term "radical" to indicate the root itself, as when we speak of solving algebraic equations by radicals.

The expression &NA; = P is called the radical expression, where n is the indicated root index, R is a real number and P is the nth root of number R such that Pⁿ = R.

The most commonly encountered radicals are the square root and the cube root. We have already discussed the square root. A bare radical sign with no indicated root index shown is understood to indicate the square root.

The cube root is the number P that solves the equation Pⁿ = R. For example, the cube root of 8, is 2.

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.

Multiplication of radicals

The product of two radicals with same index n can be found by multiplying the radicands and placing the result under the same radical. For example, √9 × √25 = √(9×25) = √225 = 15, which is equal to 3 × 5 = √9 × √25. Similarly, radicals with the same index sign can be divided by placing the quotient of the radicands under the same radical, then taking the appropriate root.

The radical of a radical can be calculated by multiplying the indexes, and placing the radicand under the appropriate radical sign. For instance, &NA; = &NA; = 2.