It is evident that, as the root, which is now 1,9129311827723, contains thirteen decimals, its cube should, according to the reasoning pursued above, differ from 7, by less than a unit of the twelfth decimal order—that is, by less than a trillionth of a unit-which it does in effect, the actual number being 6,999999999999021854419905611437426582067.

The scholar will see that, by again taking the arith. comp., and proceeding as above, he could now, by division, find twelve more figures of the root, and so on, to any degree of approximation and that thus, though we can continually approach to the number 7, we never can obtain its exact root.

385. As the cube of a unit is a unit, (170,) the cube root of a fraction must be a fraction: for, if its root could be a unit, or more, the cube of this root-that is to say, the fraction itself, would be a unit or more, which is absurd. Now, as the fraction, which is the root, is three times factor in the cube, its numerator is (206) three times factor in the numerator of the cube, and its denominator three times factor in the denominator of the cube. Therefore, to find the cube root of a fraction, when both terms are perfect cubes, we have only to find the cube root of each term, and place that of the numerator over that of the denominator. But, as we seldom find both terms perfect cubes, for a general rule, we reduce the fraction to a decimal, and find the cube root of its decimal value, by the methods already given. Or, we multiply the numerator by the square of the denominator, and divide the cube root of the product by the denominator. This is done upon the supposition that we have multiplied both terms by the square of the denominator, (which does not alter the value of the fraction,) and then divided the cube root of the numerator by that of the denominator, which is the denominator itself.

Also, we shall always find the cube of the root of a fraction within a certain limit, by extracting the root one place farther than that limit. This will easily be seen by referring to the reasoning applied, in the preceding article, to the limit of the cube root of 7.

Example.

Suppose we would extract the cube root of, so that the cube of the root may differ from by less than the hundredthousandth of a unit. We extract the root to six places of decimals, thus, (318)

For the cube root of 3, therefore, we have ,753947+; the cube of which is ,428570676309809123; and, as this cube agrees with the decimal value of 3, to the fifth decimal inclusive, it is evidently within the limit proposed. The student should prove the following

4. Required the cube root of 3, so that the cube of the root may differ from 3 by less than a ten-thousandth of a unit. Answer, 1,44224. 5. Required the cube root of 1729, so that the cube of the root may differ from 1729 by less than a thousandth of a unit. Answer, 12,002314.

In this example the operation may be abridged by placing 12 in the root, at once, seeing that 1729 only exceeds the cube of 12 by a unit. Also the three last figures, 314, may be

found by division.

The student should always find for himself the limit of the root, by the formula (a+a)3, and verify it by cubing the root.

6. Required the cube root of 1685176, so that the cube of the root may differ from the number by less than a hundredth of a unit. Answer, 119,0004.

The reason why this root reaches the limit with so few decimals is, that the next two figures of it are ciphers.

7. Required the cube root of 10795861, so that the cube of the root may differ from the number by less than a thousandth of a unit. Answer, 221,013648891.

Let the cube root of the following fractions be extracted so that the cube of the root of each fraction may differ from the fraction which gave it by less than the millionth of a unit.

386. As the index of the power to which a quantity is raised shows the number of times that the quantity is factor in that power, it is plain that, when the index is even, the number of factors is even; and, therefore, the power may

be considered as the square of the product arising from the involution of half the number of factors: consequently, if we take half the given index, the quotient will show to what power the quantity is reduced by taking its square root. Again, when the index is a multiple of 3, it is plain that the power is the cube of the product arising from the involution of one-third of the number of factors; consequently, if we take one-third of the given index, the result will show to what power the quantity is reduced by taking its cube root. Wherefore, in general, if we divide the index of a higher power of a quantity by 2, 3, 4, 5, &c., the result will show to what power the quantity is reduced by taking its square root, cube root, fourth root, fifth root, &c.

387. Hence, the roots of all those powers, the indices of which are powers of 2, may be found by the square root alone. Thus, to extract the fourth root, that is, to find a quantity from its fourth power, we first extract the square root, which gives the square of the quantity sought: we then extract the square root of the square, which gives the quantity itself. If we would extract the eighth root, the square root of the eighth power gives the fourth power; the square root of the fourth power gives the square, and the square root of the square, the

quantity itself.

388. In like manner, the roots of powers, the indices of which are powers of 3, may be found by the cube root alone. Thus, the cube root of the ninth power, gives the cube, and the cube root of the cube, the quantity itself. If we would extract the twenty-seventh root, the cube root of the twenty-seventh power, gives the ninth power; the cube root of the ninth power gives the cube, and the cube root of the cube, the quantity sought.

389. Again, we can, by the square and cube root combined, find the roots of those powers, in the indices of which 2 and 3 are the only factors. For example, to find the sixth root, we should extract the square root of the sixth power, which gives the cube; then the cube root of the cube, which gives the quantity sought. Or, first extract the cube root, and then the square root. For the twelfth root, we may first extract the square root, which will give the sixth power; the square root of the sixth power, will give the cube, and the cube root of the cube the quantity sought.

Extraction of the Roots of all Powers.

390. When the index of the power is prime to 2 and 3, and consequently cannot be resolved into those factors, we can no longer obtain the root merely by the square and cube roots, and must, therefore, have recourse to another method.

The scholar has already seen that we obtain the formula for extracting the square or cube root by considering the root under the form of a binomial, which binomial we square or cube accordingly. Now in raising a binomial successively to the higher powers, a law is found to exist in the formation of the terms of those powers, by means of which we obtain a general rule or formula for extracting the roots of all powers.

Let the binomial be a+b, as usual. Then if we raise a+b to any power, we shall find that the first term is a, raised to the given power, and the last term b, raised to the same power. Also, if we subtract a unit from the index of the power, the remainder will show the number of intermediate terms, which consist of ab, the index of a, decreasing, and that of b increasing, each by a unit at a time. The index of b in the second term, being a unit, is not written. The number of terms, therefore, is one greater than the index of the power. Thus, if we would raise a + b to the seventh power, we have, according to the above, the following eight terms: a7, aob, a3b3, a+b3, a3b, a2b3, ale, b7.

391. To have the number which precedes each of the intermediate terms, called the coefficient, we observe the following rule: The coefficient of the second term is the index of the power. For the coefficient of the third term, multiply the coefficient of the second term by the index of a in that term, and divide the product by 2. For the coefficient of the fourth term, multiply the coefficient of the third term by the index of a in that term, and divide by 3. In the same manner, to find any coefficient, multiply the coefficient of the preceding term by the index of a in that term, and divide by the number of terms to that place inclusive. Thus, for the coefficients of the above series of terms we have

for the second term, 7, the index of the power.

7 X 6