Power of a Number.—When the factors of a product are equal, the product is called a power of the factor.
Square of a Number.—A power is a square when it is the product of two (2) equal factors, as 7 × 7 = 49, in which 49 is the square of 7. The term square is derived from the fact that the area of a square is obtained by multiplying the length of its side by itself, or taking it twice as a factor.
Cube of a Number.—A power is a cube when it is the product of three (3) equal factors, as 5 × 5 × 5 = 125, in which 125 is the cube of 5.
The term cube is derived from the fact that the volume of a cube is obtained by multiplying the length of its side by itself and again by itself, or by taking it three times as a factor.
A product, for instance, of 4, 9, etc., equal factors would be called the 4th or the 9th, etc., power of that number.
Division.
Division.—An operation by means of which we find one of two factors of a product when that product and the other factor are given. The given product is called Dividend (D) of the division; the known factor is called the Divisor (d), and the unknown factor which is sought is called Quotient (q). We know that a quotient is seldom exact and that there is generally a Remainder (r) or Residue.
Sign of Division.—The sign of division is a small dash with a point above and one below ÷; it is read divided by, is placed after the dividend, and is followed by the divisor. For instance, to indicate the division of 72 by 8, which we know gives the quotient 9, we write 72 ÷ 8 = 9; generally D ÷ d = q.
Other Sign of Division.—In the study of fractions it is shown that a fraction expresses the quotient of its numerator by its denominator, so that the preceding identity may be written 72 8 = 9, or more generally D d = q, and another sign of division is a horizontal line separating the dividend written above it from the divisor written below it.
Proof of the Division.—We prove a division by multiplying the divisor by the quotient and adding the remainder, if there is any; the result thus obtained must equal the dividend. When there is a remainder, the formula of division is D = dq + r.