inal. Hence, from the preceding principle, we have the following rule for finding the nth root of any decimal fraction to any desired degree of accuracy.

RULE.

Annex O's to the given decima, till the number of decimal places is n times the number in the required root; extract the nth root of the result, as though it were a whole number, and point off the required number of decimal places in the root.

The rule is applicable when the fraction is a vulgar one, for we may convert it into a decimal one by known rules; it is also applicable to finding an approximate root of a whole number. It may be shown, as in square root, that the nth root of a whole number can never be a fraction; hence, if the number is not a perfect 7th power, its nth root can never be exactly found.

EXAMPLES.

Find the cube roots of the following numbers, approxi mately:

120. We have seen (Art. 99) that any monomial can be raised to any power by raising the coefficient to the required power for a new coefficient, and giving to each letter an exponent equal to its original exponent, mul tiplied by the exponent of the required power. Revers ing the process, we have a rule for extracting any root of a monomial.

RULE.

Extract the required root of the coefficient for a new coefficient; after this, write all the letters, giving to each an exponent equal to its original exponent, divided by the index of the required root..

This rule, combined with the rule for extracting any root of a fraction already given, enables us to extract any root of a monomial, whether entire or fractional.

9. Find the cube root of 343x-3y-6,

10. Find the fifth root of x-10y-205.

Ans. 7x-y-2.

Ans. x-2y-4z.

Since the square of a is a2, and the square of a is also a2, it follows that a2 has two square roots, + a and a. Further, since every even power of a positive quantity is equal to the same power of that quantity taken with a negative sign, it follows that every positive quantity has two square roots, two fourth roots, two sixth roots, &c., which are equal numerically, but have contrary signs. Thus,

√25a2b2 = ± 5ab; †/16a1b8 = ± 2ab2; §3⁄4/a12b¤ = ± a2b.

It is impossible that any even power of a quantity, either positive or negative, should be a negative quan. tity. Hence, it is equally impossible to extract any even root of a negative quantity. An indicated even root of a negative quantity, is called an imaginary quantity. Thus, 4, - a2, & b2 are imaginary quanti ties.

Every odd power of a negative quantity is itself nega tive, and consequently every odd root of a negative quantity must be negative. Every odd root of a posi sive quantity must of course be positive. Hence, the

RULE.

Every even root of a positive quantity must have the double sign, ±; every odd root of any quantity must have the sign of the quantity.

In the preceding examples, only the numerical value of the roots have been considered; hereafter, the proper signs will be prefixed to the results.

EXAMPLES.

11. Find the cube root of

27a3x-3y6.
Ans.

64

3ax-1y2. 8

12. Find the square root of ab. Ans. ± a2b2.

121. To deduce a rule for extracting the square root of a polynomial, let us suppose the root to be known, and to be arranged with respect to some letter. We may regard the root as composed of the first term plus the sum of all the other terms. Hence, its square, which is the given polynomial, will be made up of the square of the first term, plus twice the product of the first term by the sum of all the other terms, plus the square of the sum of all the other terms (Art. 106). Now the square of the first term of the root must be of a higher degree with respect to the leading letter than any other term. Hence, if we arrange the given polynomial with respect to any letter, the square root of the first term will be the first term of the required square root. If the square of this term of the root be subtracted from the given polynomial, and the first term of the remainder be divided by twice the first term of the root, the quotient will be the second term of the root. The remainder of the process of finding the square root, is entirely analogous to that already explained for finding the square root of a whole number, and will be best understood from an example. Let it be required to extract the square root of 9x4 + 12x3 + 28x2 + 16x + 16.