This section is from "The American Cyclopaedia", by George Ripley And Charles A. Dana. Also available from Amazon: The New American Cyclopędia. 16 volumes complete..

**Root**, in mathematics, a term used in two different though related senses. I. In arithmetic a root is the inverse of a power; thus 16 is the fourth power of 2, and 2 is the fourth root of 16; 9 is the second power, or, as it is usually called, the square of 3, and 3 is the second or square root of 9. It will thus be seen that a root of a given number is a number which being taken a certain number of times as a factor will produce the given number. An arithmetical root of a number is indicated by the sign √ with the number placed after it, and the number indicating the degree of the root placed above and before it; thus
16 is read "the fourth root of 16." The sign is a modification of the letter r, which was formerly used for this purpose. The second or square root is indicated by the sign alone, the figure 2 being omitted; thus √9 means the same as
9. The first root of a number is the number itself, and therefore needs no sign. In the best modern works on algebra the sign √ is strictly limited to the designation of the arithmetical root of a quantity. II. In algebra the term root is used to denote any value of the unknown quantity in an equation, which being substituted for that quantity will satisfy the equation; thus the equation x4-7x3+ 27x2-47x+26=0 can be satisfied by substituting for x any one of four different values, two of which are real, viz., 1 and 2, and two are imaginary, viz. (2+3√-1) and (2-3√-1). Any given number has only one arithmetical root of a given degree, but it has as many algebraical roots as there are units in the number denoting the degree of the root; thus 3 is the only arithmetical square root of 9, but algebraically 9 has two square roots, +3 and -3, because either of these will satisfy the equation x2=9. The arithmetical root is at the same time one of the algebraical roots.

Algebraical roots are designated by means of fractional exponents (see Exponent); thus 91/2 may be considered as representing either +3 or -3, and 11/4 means either 1 (its arithmetical value) or -1 or √-1 or -√-1, either of which will satisfy the equation x4=1. "When the numerator of the fractional exponent is some number other than 1, it indicates that the root expressed by the denominator is to be raised to the power expressed by the numerator; thus a3/5 means the third power of any one of the five fifth roots of a.

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