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 II M E T I C K. Chap. XII. Extracti6n of Roots. If unity be multiplied continually by any given number, the produ&s thence afifing are called powers of that number; and the given number is called the root, or firjl power. Thus, if 2 be the given number, then 1X2=2 is the 'in root or firft power ; and 2X2=4 *s t^e fqnaie or fecond 8333 power; and 4X2=8 is the cube or third power; and 41^666 8X2=16 is the biquadrate or fourth power; and 16X 2=32 is the furfolid or fifth power; and 32X2=64 is 14.283)42.8 5bo( the fixth power, or cube fquared, <&c. 1428 42.850 The natural numbers, 1, 2, 3, <bc. are fometimes placed over thefe powers, denoting the number of Mul12855)38565-6(3 Jnf. tiplications ufed in producing them, of (bowing what 38565 powers they are ; and are called indices or exponents, as in the following fcheme. Rule of Three Inverfe. Indices, o, 1, 2, 3, 41 5, 6, 7, he. i, 2, 4, 8, 16, 32, 64, 128, tec. Examp. ' If you borrow L. 64 for 8 months, what ThePowers, raifing any root or number given to any power fum lent ? for 12 months, or a year, will recjuite the fa- required, is called involution; and is performed by mulvour tiplying the given root into unity continually, as taught r. l. r. above. But the finding the root of a given power is Vulgar ftate 1 : 64 :: called or ext raft ion of roots. Decimal ftate 1 : 64 :: & If the root of any power not exceeding the feventh 6 power, be a fingle digit, it may be obtained by infpecL. s. d. tion, from the following table of powers. 9)384(42.^=42 13 4 T A B L E. 36; 24!8 Or thus:

A R I T 420 lb. oz. diu. L. t. Vulgar Rate 14 ? 8 : 514 4 :: 083 Decimal Rate 14.283 • SH‘2 5:14.2

60 3)64(21-3' 54 ' 2i£ Compound Rule of Three. Examp. What is the intereft of L. 75 : 10 : 4 for 8 months, at the rate of 5 per cent, per annum ? M. L. L. L. s. d. M. Vulgar ftate 12 X 100 : 5 :: 75 10 4X8 Decimal ftate 12 X 100 : 5 :: 75.51^ X 8 12 8 1200 : 5 :: 604.133
 * 6 The fame 42.^ as before.

I. Extraftion of the Square Root. Rule I. Divide the given number into periods of two figures, beginning at the right hand in integers, i2|oo) 3020.666( and pointing toward the left. But in decimals, begin at d. I. the place of hundreds, and point toward the right. E10 4 2.517^ very period will give one figure in the root. II. Find by the table of powers, or by trial, the The Ample feparate operations of the fame example neareft leffer root of the left-hand period, place the fifollow. gure fo found in the quot, fubtradl its fquare from the L. L. L. M. L. M. faid period, and to the remainder bring down the next 100: 5" 7S-Si6 12:3.77583-8 8 period for a dividual or refolvend. 5 III. Double the quot for the firft part of the diviinquire bow often this firft part is contained in the I2)30.20$66( for; j1oo)377-583’( whole refolvend, excluding the units place; and place Z.WiAnf. the figure denoting the anfwer both in the quot and theon 3-7758^