XXXVI. Of Powers and Roots in General. 183 frequently necessary to find the roots of other powers, as well as of the second and third, and of literal, as well as of numeral quantities. Preparatory to this, it is necessary to attend a little more particularly to the formation of powers. The second power of a is a x a = a*. If a quantity as a is multiplied into itself until it enters m times as a factor, it is said to be raised to the oth power, and is expressed am This is done by m-1 multiplications ; for one multiplication as a x a produces a’ the second power, two multiplications produce the third power, &c. We have seen above Art. X. that when the quantities to be multiplied are alike, the multiplication is performed by adding exponents. By this principle it is easy to find any power of a quantity which is already a power. Thus The second power of a® is a xao = 243 = a*. power of a' is á xa x a* = a +2+ = a®. amtmtm The mth power of a’ is a * a? x ax a X =a2+2+2+2+......, until a’ is taken m times as a factor, that is, until the exponent 2 has been taken m times. Hence it is expressed a2m The nth power of am is an X am X am until m is taken n times, and the power is expressed amn. N. B. The dots ..... in the two last examples are used to express the continuation of the multiplication or addition, because it cannot come to an end until in in the first case, and n in the second, receive a determinate value. In looking over the above examples we observe ; 1st. That the second power of as is the same as the third power of a’, and so of all others. =a?m. =a3m. ....= amtmtmt. 2. That in finding a power of a letter the exponent is added until it is taken as many times as there are units in the exponent of the required power. Hence any quantity may be raised to any power by, multiplying its exponent by the exponent of the power to which it is to be raised. The 5th power of a’ is a3X5 = alb. The 3d power of a' is a7x3 = a*, &c. . The power of a product is the same as the product of that power of all its factors. The 2d power of 3 ab is 3 ab x 3 ab = 9 aob. Hence, when a quantity consists of several letters, it may be raised to any power by multiplying the exponents of each letter by the exponent of the power required ; and if the quantity has a numeral coefficient, that must be raised to the power required. The powers of a fraction are found by raising both numerator and denominator to the power required ; for that is equivalent to the continued multiplication of the fraction by itself. 1 What is the 5th power of 3 a* b* in ? 2 ac 2 What is the 3d power ? Powers of compound quantities are found like those of simple quantities, by the continued multiplication of the quantity into itself. The second power is found by multiplying the quantity once by itself. The third power is found by two multiplications, &c. The powers of compound quantities are expressed by enclosing the quantities in a parenthesis, or hy drawing a vinculum over them, and giving them the exponent of the power. The third power of a + 2 b c is expressed (a + 2b-c); or & + 26 d. The powers are found by multiplication as follows: a + 4 ab + 46 — 2 ac-4bc+c=(a +26 - 0)* a +2b-C ai + 4 a 6 + 4 a b— 2 a' c-4abct ac - a'c -4abc-46*c+2 ac* +466_o a + 6 a b + 12 a b + 863 — 3 ac— 12 abc- 126*c + 3a c + 6 6 c* — c = (a + 2b-c)*. If the third power be multiplied by a + 2b-c, it will produce the fourth power. 3. What is the second power of 3c+2 d? 4. What is the third power of 4 a - b c? 5. What is the fifth power of a -6? 6. What is the fourth power of 2 a* c-? In practice it is generally more convenient to express the powers of compound quantities, than actually to find them by multiplication. And operations may frequently be more easily performed on them when they are only expressed. (a +5)* (a + b)*= (a + b)4= (a + b) That is, when one power of a compound quantity is to be multiplied by any power of the same quantity, it may be expressed by adding the exponents, in the same manner as simple quantities. The 2d power of (a + b)' is (a +b)* (a + b)" (a +b)343 = (a + b)*x? = (a + b). The 3d power of (2 a (2 a—d)4+4+4 (2a —d)-x} = (2a - d)'?. That is, any quantity, which is already a power of a compound quantity, may be raised to any power by multiplying its exponent by the exponent of the power to which it is to be raised. 7. Express the 2d power of (3 b-c). -d)* is 8. Express the 3d power of ( ac+2 d). 9. Express the 7th power of (2 a — 4 c9). Division may also be performed by subtracting the exponents as in simple quantities. (3a - b)divided by (3a — b) is (3a—b)–3 = (3 a—b) 10. Divide (7 m +2 c)' by (7 m + 2 c)". If (a + b) is to be multiplied by any quantity c, it may be expressed thus : c (a + b). But in order to perform the operation, the 2d power of a + b must first be found. c(n +5)*.= c(a + 2ab + b*) = a*c+2 abc + bc If the operation were performeti previously, a very erroneous result would be obtained; for c (a + b) is very different from (ac + bc)'. The value of the latter expression is ac + 2 ab c* + b2c. 11. What is the value of 2 (a + 3b) developed as above? 12. What is the value of 3 bc (2a ---c)? 13. What is the value of (a + 3c*) (3a —26)? 14. What is the value of (2a-6)* (a + bc)'? XXXVI. Of Powers and Roots in General. 187 We have had occasion in the preceding pages to return from the second and third powers to their roots. We have shown how this can be done in numeral quantities; it remains to be shown how it may be effected in literal quantities. It is frequently necessary to find the roots of other powers as well as of the second and third. The power of a literal quantity, we have just seen, is found by multiplying its exponent by the exponent of the power to which it is to be raised. The second power of a® is a X'= a'; consequently the second root of ac is aš = a*. The third power of am is am ; hence the third root of asm 3m must be a =a". m The second root of am, then must be az. In general, the root of a literal quantity may be found by divid. ring its exponent by the number expressing the root ;, that is, by dividing by 2 for the second root, by 3 for the third root, &c. This is the reverse of the method of finding powers. It was shown above, that any power of a quantity consisting of several factors is the same as the product of the powers of the several factors. From this it follows, that any root of a quantity consisting of several factors is the same as the product of the roots of all the factors. The third power of a' b c is a® 6* co; the third root of a® 63 ce must therefore be a b c. Numeral coefficients are, factors, and in finding powers they are raised to the power; consequently in finding roots, the root of the coefficient must be taken. The 2nd root of 16 a* b* is 4 a® b. Proof 4 a 6 x 4 a b = 16 a*b*. |