ROOT

, in Arithmetic and Algebra, denotes a quantity which being multiplied by itself produces some higher power; or a quantity considered as the basis or foundation of a higher power, out of which this arises and grows, like as a plant from its Root.

In the involution of powers, from a given Root, the Root is also called the first power; when this is once multiplied by itself, it produces the square or second power; this multiplied by the Root again, makes the cube or 3d power; and so on. And hence the Roots also come to be denominated the square-Root, or cube-Root, or 2d Root, or 3d Root, &c, according as the given power or quantity is considered as the square, or cube, or 2d power, or 3d power, &c. Thus, 2 is the square-Root or 2d Root of 4, and the cube-Root or 3d Root of 8, and the 4th Root of 16, &c.

Hence, the square-Root is the mean proportional between 1 and the square or given power; and the cube-Root is the first of two mean proportionals between 1 and the given cube; and so on.

To Extract the Root of a given number or power. This is the same thing as to find a number or quantity, which being multiplied the proper number of times, will produce the given number or power. So, to find the cube Root of 8, is finding the number 2, which multiplied twice by itself produces the given number 8.

For the usual methods of extracting the Roots of Numbers, see the common treatises on Arithmetic.

A Root, of any power, that consists of two parts, is called a binomial Root; as 12 or 10 + 2. If it consist of three parts, it is a trinomial Root; as 126 or 100 + 20 + 6. And so on.

The extraction of the Roots of algebraic quantities, is also performed after the same manner as that of numbers; as may be seen in any treatise on algebra. See also the article Extraction of Roots.

A general method for all Roots, is also by Newton's binomial theorem. See Binomial Theorem.

Finite approximating rules for the extraction of Roots have also been given by several authors, as Raphson, De Lagney, Halley, &c. See the articles APPROXIMATION and Extraction. See also Newton's Universal Arith. the Appendix; Philos. Trans. numb. 210, or Abridg. vol. 1, pa. 81; Maclaurin's Alg. pa. 242; Simpson's Alg. pa. 155; or his Essays, pa. 82, or his Dissertations, pa. 102, or his Select Exerc. pa. 215: where various general theorems for approximating to the Roots of pure powers are given. See also Equation and Reduction of Equations, APPROXIMATION, and Converging.

But the most commodious and general rule of any, for such approximations, I believe, is that which has been invented by myself, and explained in my Tracts, vol. 1, pa. 49: which theorem is this; . That is, having to extract the nth Root of the given number N; take aⁿ the nearest rational power to that given quantity N, whether greater or less, its Root of the same kind being a; then the required Root, or √ⁿN, will be as is expressed in this formula above; or the same expressed in a proportion will be thus: the Root sought very nearly. Which rule includes all the particular rational formulas of De Lagney, and Halley, which were separately investigated by them; and yet this general formula is perfectly simple and easy to apply, and more easily kept in mind than any one of the said particular formulas.

Ex. Suppose it be required to double the cube, or to extract the cube Root of the number 2.

Here N = 2, n = 3, the nearest Root a = 1, also a³ = 1; hence, for the cube Root the formula becomes .

But ; therefore as 4 : 5 :: 1 : 5/4 = 1.25 = the Root nearly by a first approximation.

Again, for a second approximation, take a = 5/4, and consequently ; therefore as 378 : 381, or as 126 : 127 :: 5/4 : 635/504 = 1.259921 &c, for the required cube Root of 2, which is true even in the last place of decimals.

Root of an Equation, denotes the value of the unknown quantity in an equation; which is such a | quantity, as being substituted instead of that unknown letter, into the equation, shall make all the terms to vanish, or both sides equal to each other. Thus, of the equation , the Root or value of x is 3, because substituting 3 for x, makes it become 9 + 5 = 14. And the Root of the equation is 4, because 2 X 4² = 32. Also the Root of the equation .

For the Nature of Roots, and for extracting the several Roots of equations, see Equation.

Every equation has as many Roots, or values of the unknown quantity, as are the dimensions or highest power in it. As a simple equation one Root, a quadratic two, a cubic three, and so on.

Roots are positive or negative, real or imaginary, rational or radical, &c. See Equation.

Cubic Root. This is threefold, even for a simple cubic. So the cube Root of a³, is either . And even the cube Root of 1 itself is either .

Real and Imaginary Roots. The odd Roots, as the 3d, 5th, 7th, &c Roots, of all real quantities, whether positive or negative, are real, and are respectively positive or negative. So the cube Root of a³ is a, and of - a³ is - a.

But the even Roots, as the 2d, 4th, 6th, &c, are only real when the quantity is positive; being imaginary or impossible when the quantity is negative. So the square Root of a² is a, which is real; but the square Root of - a², that is, √(- a²), is imaginary or impossible; because there is no quantity, neither + a nor - a, which by squaring will make the given negative square - a².

THE following Table of Roots, Squares, and Cubes, is very useful in many calculations, and will serve to find square-Roots and cube Roots, as well as square and cubic powers. The Table consists of three columns: in the first column are the series of common numbers, or Roots, 1, 2, 3, 4, 5, 6, &c; in the second column are the squares, and in the third column the cubes of the same. For example, to find the square or the cube of the number or Root 49. Finding this number 49 in the first column; upon the same line with it, stands its square 2401 in the second column, and its cube 117649 in the third column.

Again, to find the square Root of the number 700. Near the beginning of the Table, it appears that the next less and greater tabular squares are 676 and 729, whose Roots are 26 and 27, and therefore the square Root of 700 is between 26 and 27. But a little further on, viz, among the hundreds, it appears that the required Root lies between 26.4 and 26.5, the tabular squares of these being 696.96 and 702.25, cutting off the proper part of the figures for decimals. Take the difference between the less square 696.96 and the given number 700, which gives 3.04, and divide the half of it, viz 1.52, by the less given tabular Root, viz 26.4, and the quotient 575 gives as many more figures of the Root, to be joined to the first three, and thus making the Root equal to 26.4575, which is true in all its places.

Also to find the cube Root of the number 7000; near the beginning of the Table, among the tens, it appears that the cube Root of this number is between 19 and 20; but farther on, among the hundreds, it appears that it lies between 19.1 and 19.2, allowing for the proper number of integers. But if more figures are required; from the given number 7000 take the next less tabular one, or the cube of 19.1, viz 6967871, and there remains 32.129, the 3d part of which, or 10.730, divide by the square of 19.1, viz 364.81, found on the same line, and the quotient 293 is the next three figures of the Root, and therefore the whole cubic Root is 19.1293, which is true in all its figures.—The Table follows. |

The following is another Table of the Square Roots of the first 1000 Numbers to 10 places of decimal figures beside the integers, which needs no farther explanation, as Numbers stand always in the first column, and their Square Roots in the next.