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. Mathematicians indicate a number raised to the n th power by writing xn. This expression indicates that one is 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 one is seeking the square root of x2, which equals x× x, then one is seeking the variable that, when multiplied by itself, is equal to x2—namely, x. That is to say, = 32 = 3×3. Similarly, if one is looking for the fourth root of x4,then one is 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 [nth root] √ 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 one speaks of solving algebraic equations by radicals.
The expression [nthroot] = P is called the radical expression, where n is the indicated root index, R is a real number, and P is the n th root of number R such that Pn = R.
The most commonly encountered radicals are the square root and the cube root. The square root has already been discussed. 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: = 3, [fourth root of 16]=±2.
Because the root P of [nth root]R 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, [cube root of 8]= 2(23 = 2× 2× 2= 4× 2= 8), but [cube root of–8]=–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
Imaginary number— A number multiplied by the imaginary unit i, which is equal to
Index— Order of the root. For example, the index of a cube root is 3.
Radicand— The number under the radical sign.
Root sign— A symbol that indicates the radical or root operation.
unit or 2 i=–1. Then, The imaginary unit is a very useful concept in certain types of calculus and complex analysis.
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, which is equal to 3 × 5 = × . 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, [cube root of the square root of 64]= [sixth root of 64]= 2.