« ΠροηγούμενηΣυνέχεια »
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 544. The remainder must contain 5 a* x + 10 a® * +, &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 544,35324. 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 third 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 quantity consisting of several factors, is the same as the product of the roots of the several factors. . Hence (8° 69)* = (20)*(69)* = a* b.
(29)} = (a)7. (a)} = (a). (a)(a)*
= at. at. at = a. at = ah+ = al. We see that the same expression may be written in a great many different forms. The most remarkable of the above are
al = alt = a1 + 1 = a. at 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 conve. nient time.
The quantity (72 a* b* c)* may be resolved into factors thus. (2 X 36 ao abbc)* = (36 a*b**. (2 abc).
The root of the first factor 36 a*b* can be found exactly, and the expression becomes
6 a b* (2 a bc) 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, (72)+ ax b4 c4 = (36.2)14727,= 6.25 a atze68cm
= 6 a b* (2 a bc)
5. Redure (135 *w_- 108 are ca 3
to its simplest form.
3. Reduce (
Sometimes it is convenient to multiply a root by another quantity, or one root by another.
If it is required to multiply ( ab, it may be expressed thus : a b (3 a*b)*. But if it is required actually to unite them, ab must first be raised to the second power, and the pro
duct becomes (3 04 63). This will appear more plain in the following manner,
(3 ab)= 34 a bit This multiplied by ab is .
35 a 64 x ab = 3+ 40214 = 35 (2]} = (3 a4 bayt. If instead of enclosing the quantity in the parenthesis and writing the exponent of the root over it, we divide the exponent of all the factors by the exponent of the root, all the operations will be very simple. Let at be multiplied by at...
at xut = ał+1 = a.
at xal = 7+1 = a* = a.. po 33 x aš bi -- a3+33+1 = aš . That is, multiplication is performed on similar quantities by adding the exponents, as when the exponents are whole numbers. In like manner division is performed by subtracting the exponents.
It must be observed that as may be read, the third root of the second power of a, or the second power of the third root of a. For the 3d root of a? is al; and
The 3d power of at is
aš xaš xaš = a3+1+1 = a*X3 = a8. That is, a power of a root may be found by multiplying the fractional exponent by the exponent of the power. Consequently a root of a root may be found by dividing the fractional index by the exponent of the root. In multiplying and dividing the fractional exponents, we must apply the same rules that we apply to common fractions.
The 3d root of aš is aš.
If the numerator and denominator both be multiplied or divided by the same number, the value of the quantity will not be altered; for that is the same as raising it to a power, and then extracting the root.
If it is required to multiply ał by a, the fractions may be reduced to a common denominator and added : thus,
al x at = at xaš = aš = ++ = a at. The same may be done in division and the exponents subtracted.
In fact, quantities with fractional exponents are subject to precisely the same rules, as when the exponents are whole numbers; but the rules must be applied as to fractions. The fractions may be reduced to decimals without altering the value ; thus at = a14 = 41.25 = a x 2.25 = x 20.3 x 2.06
· =axaro xaroo.