Those roots which only approximate are called Surd-roots; but those which can be found, quite exactly, are called Rational-roots. Thus, the square root of 3 is a surd root, but the square root of 4 is a rational root, being equal to 2; also the cube root of 8 is rational, being equal to 2, but the cube root of 9 is surd, or irrational. Roots are sometimes denoted by writing the character √ before the power with the index of the root against it. Thus, the 3rd, or cube root of 20 is expressed by ∛20. When the power is expressed by several numbers with the sign + or - between them, a line is drawn from the top of the sign over all the parts of it; thus the cube (or third) root of 45 - 12 is ∛45 - 12 or thus ∛(45 - 12).
TO EXTRACT THE SQUARE ROOT.
Rule.—Divide the given number into periods of two figures each, by setting a point over the place of units, and another over the place of hundreds, and so on over every second figure, both to the left hand in integers, and right hand in decimals. Find the greatest square in the first period on the left hand, and set its root on the right hand of the given number, after the manner of the quotient figure in division. Subtract the square thus found from the said period, and to the remainder annex the two figures of the next following period for a dividend. Double[45] the root above-mentioned for a divisor, and find how often it is contained in the said dividend, exclusive of its right-hand figure; and set that quotient figure both in the quotient, and divisor. Multiply the whole augmented divisor by this last quotient figure, and subtract the product from the said dividend, bringing down to it the next period of the given number, for a new dividend. Repeat the same process over again—viz., find another new divisor, by doubling all the figures now found in the root; from which, and the last dividend find the next figure of the root as before; and so on through all the periods to the last.
To extract the square root of a fraction, or mixed number.
Reduce the fraction to a decimal, and extract its root.
Mixed numbers may be either reduced to improper fractions, and the root extracted; or the fraction may be reduced to a decimal, then joined to the integer, and the root of the whole extracted.
Example.—To find the square root of 29506624.
| 29506624 | ( 5432 The Root. | |
| 25 | ||
| 104 | 450 | |
| 4 | 416 | |
| 1083 | 3466 | |
| 3 | 3249 | |
| 10862 | 21724 | |
| 2 | 21724 |
TO EXTRACT THE CUBE ROOT.
Rule 1.—By trials, or by the table of roots (vide page [280]), take the nearest rational cube to the given number, whether it be greater, or less, and call it the assumed cube.