The degree of the power to which a number is to be raised, is indicated by writing to the right of the given number, and a little above, a small figure expressing the degree. Thus, 238 is the 3d power of 2. This figure is called the index or exponent of the power. = 42 = 16 is the 2d power of 4. 26 = 64 is the 6th power of 2. 134 = 28561 is the 4th power of 13. (53.) When two or more powers are multiplied together, their product is that power whose exponent is the sum of the exponents of the powers multiplied; or, the multiplication of the powers, answers to the addition of the exponents, as will appear from the following example of the powers of 2: By multiplying the 2d and 4th, we have 4 × 16 = 64, and adding the exponents from the second line, 22×24 = 26 = 64. We have also, (54.) Multiply the given number by itself a number of times less 1, than the exponent of the required power; or take two or more powers of the given number the sum of whose exponents is equal to the exponent of the required power, and multiply them together successively. EXAMPLES. 1. Involve 67 to the 2d power. 67×67=4489, the answer. 2. What is the 3d power of 93? 93 × 93 × 93 = 804357, the answer. 3. What is the 8th power of 4? 4x4x4x4x4x4x4x4 = 65536 ; or, 4' x 4' = 4; that is, 256 x 256 = 65536; or, 42 × 43 × 43 = 4; that is, 16 x64×64 = 65536, the ans. 4. What is the square of 4,16? 5. What is the 3d power of 3,5? 6. What is the 4th power of 0,05? 7. What is the 2d power of 899? 8. What is the 6th power of 26? 9. What is the 2d power of ? 10. What is the 4th power of? 11. What is the 3d power of ? 12. What is the 5th power of 2? Ans. 17,3056. Ans. 42,875. Ans. 0,00000625. Ans. 808201. Ans. 308915776. =, the answer. ××× = 1, the answer. 15. What is the 4th power of 23? Ans. 1, or, 201. 6 Ans. 40, or, 50%. EVOLUTION. (55.) Evolution is the reverse of involution. It is the extracting or finding the roots of any given powers. The root of any number, is such a number as multiplied into itself a certain number of times, will produce that number; thus, = 5 is the square or 2d root of 25, since 52, or, 5 × 5 4 is the cube or 3d root of 64, since 43, or, 4×4×4 The root of a number is denoted by writing the character ✓ before the number, and the degree of the root is shown by writing a small figure, expressing the degree directly over the 2 sign; v4 or 3 4 is the square root of 4; 4 is the cube 4 or 3d root of 4; 4, the 4th root of 4, &c. EXTRACTION OF THE SQUARE ROOT.* RULE. (56.) Divide the given number into periods of two figures each, by placing a dot over the unit figure, and another over every second figure to the left. Find the greatest square in the left hand period, and set its root on the right, after the manner of a quotient in division. Subtract the square, thus found, from the said period, and to the remainder annex the figures of the next period for a dividend. Double the root, already found, for a divisor, and find how many times it is contained in the dividend, exclusive of the right hand figure, and place that quotient figure both in the quotient and divisor. Multiply the divisor, thus increased, by this quotient figure, subtract the product from the dividend, and annex the next period to the remainder, for a new dividend. Find a new divisor, by doubling all the figures now found in the root, and proceed as before through all the periods. * The demonstration of the rules for extracting roots will be given in Algebra. EXAMPLES. 1. What is the square root of 29506624 ? 29506624(5432, the root. 25 104(450 1083)3466 3249 10862)21724 NOTE I.-There are many numbers the square root of which cannot be extracted exactly; but we can arrive very nearly at the root by continuing the operation in decimals; and this is done by adding periods of two ciphers after the whole numbers are all brought down. 2. For example, let it be required to find the square root of 312. 312(17,6635, the root. 1 27)212 346)2300 3526)22400 35323)124400 353265)1843100 76775 NOTE II.-If the number consist of a whole number and a decimal, it is necessary to divide the decimal part into periods, beginning at the decimal point, and counting two figures to the right for each period; and if the right hand period of the decimal should contain but one figure, a cipher must be annexed to it. 4. Let it be required to find the square root of 220,758 220,7580(14,85 1 24)120 96 288)2475 2965)17180 14825 2355 5. What is the square root of 2025? 6. What is the square root of 17,3056? Ans. 45. Ans. 4,16 |