XXXIX.

Extraction of the Roots of Numeral Quantities of any Degree.

By the above expression of the several powers, we may extract any root of a numeral quantity. Let us take a particular example.

What is the 5th root of 5,443,532,400,000?

In the first place we observe that the 5th power of 10 is 100000, and the 5th power of 100 is 10000000000. Therefore if the root contains a figure in the ten's place, it must be sought among the figures at the left of the first five places counting from the right. Also if the root contains a figure in the hundred's place, it must be sought at the left of the first ten figures. This shows that the number may be divided into periods of five figures each, beginning at the right. The number so prepared will stand

In the first place I find the greatest 5th power in 544. It is 243, the root of which is 3. I write 3 in the root, and subtract 243, the 5th power of 3, from 244. The remainder must contain 5 a x + 10 a3 x2 + &c. The 3, that part of the root already found, and

which, by the number of periods, must be 300, answers to a in the formula. 5 a, that is, five times the fourth power of 300 will form only an approximate divisor, since the remainder consists of several terms besides 5 a+ x; still it will enable us to judge very nearly, and we shall find the right number after one or two trials. As the fourth power of 30 will have no significant figure below 10000, (we may consider 3 to be in the ten's place, = with regard to the next figure to be found), we may bring down only one figure of the next period to the remainder for the dividend, and use 5 times the fourth power of 3 for the divisor. The dividend is 3013 and the divisor 405. The dividend contains the divisor at least 6 times, but probably 6 is too large for the root. Try 5. This gives for the first two figures 35. Raise 35 to the 5th power and see if it is equal to 5441,25324, It will exceed it. Therefore try 4. The fifth power of 34 is 544,35324. Hence 34 is right. Subtract this from the number, there is no remainder. There is still another period, but it contains no significant figure, therefore the next figure is 0, and the root is 340. The 5th power of 340 is 5,443,532,400,000. If there had been a remainder after subtracting the 5th power of 34, it would have been necessary to bring down the next figure of the number to it to form a dividend, and then to divide it by 5 times the 4th power of 34; and to proceed in all respects as before.

The process of extracting roots above the second is very tedious. A method of doing it by logarithms will hereafter be shown, by which it may be much more expeditiously performed.

Examples.

1. What is the 5th root of 15937022465957 ?

2. What is the 4th root of 36469158961 ?

For this, the fourth root may be extracted directly, or it may be done by two extractions of the second root. Let the learner do it both ways.

3. What is the 6th root of 481890304 ?

This may be done by extracting the 6th root directly, or by extracting first the second and then the 3d root. Let it be done both ways.

4. What is the 7th root of 13492928512 ?

XL. Fractional Exponents and Irrational Quantities.

The method explained above, Art. XXXVI, for extracting the roots of literal quantities, gives rise to fractional exponents, when they cannot be exactly divided by the number expressing the root. Since quantities of this kind frequently occur, mathematicians have invented methods of performing the different operations upon them in the same manner as if the roots could be found exactly; and thus putting off the actual extracting of the root until the last, if it happens to be most convenient. The expressions also may often be reduced to others much more simple, and whose roots may be more easily found.

It has been already observed that the root of a quan

tity consisting of several factors, is the same as the product of the roots of the several factors.

We see that the same expression may be written in a great many different forms. The most remarkable of

the above are,

= α

14 1+1

=α = a.a

On this principle we may actually take the root of a part of the factors of a quantity when they have roots, and leave the roots of the others to be taken by approximation at a convenient time.

1

The quantity (72 a b c) may be resolved inte factors thus.

a2 b4)3.(2 a b c)3.

¡¡(2 × 36 aa a b• b c)3 — (36 a2 b

: =

The root of the first factor 36 a2 b4 can be found exactly, and the expression becomes

6 a b2 (2 a b c ) 3.

This expression is much more simple than the other, for now it is necessary to find the root of only 2 abc.

The expression might have been put in this form,

1. Reduce (16 aa b♦)3 to its simplest form.

Ans. 2 a b (2a2 b)3.

2. Reduce (54 a x7) to its simplest form.

1

4. Reduce (16 a3 b3 + 32 a3 b3 m) to its simplest

form.

(16 a3 b* + 32 a2 b3 m)3 = (16 a3 b2)3 (ab3 + 2 bm)

+2b

Sometimes it is convenient to multiply a root by another quantity, or one root by another.