INVOLUTION. 54. INVOLUTION is the raising of any given quantity to any proposed power; such as the square, cube, &c. If a quantity be continually multiplied by itself, it is said to be involved, or raised; and the power to which it is raised is expressed by the number of times the quantity has been employed in the multiplication; thus aXa, or a2, is called the second power of a; aXaXa, or a3, the cube, or third power; aXa....(n), or a", the nth power. RULE. When the quantity or root has no index, its power will be represented by placing the index of the required power above it; thus the fourth power of x is x1; the fourth power of x+y is (x+y). If the quantity proposed be a compound one, the involution may either be represented by the proper index, or it may actually take place. If the quantity to be involved be negative, the signs of the even powers will be positive, and the signs of the odd powers negative; for aX-a—a2; —aX-ax—a——a3. If the quantity be a fraction, raise both the numerator and denominator to the same power. When the quantity to be raised is itself a power, multiply the index of the quantity by the index of the power; thus the cube of a2 is ao a2x3 or generally let m or n represent any powers whatever; then 2", raised to the nth power is xx", or ""; and —x", raised to any power, n, will give +, plus, or —, minus, according as n is an odd or even number. If n be used for an uneven number, the sign will be +, if odd —; then-x" to the nth power, will be represented by Roots and Powers of Numbers. mn Xn 4th power 1 16 81 256 625 1296 2401 4096 6561 10000 5th 243 power 1 32 243 1024 3125 7776 16807 32768 59049 100000 The operation is performed in the same manner for simple algebraic quantities, except that in this case it must be observed, that the powers of negative quantities are alternately + and —; the even powers being positive, and the odd powers negative. Thus the square of +2a is +2aX+2a, or 4a2; the square of -2a is-2aX-2a, or +4a2; but the cube of 2aX-2a=+4a3X-2a-8a3. Upon this principle the powers of the several roots in the following table are calculated. Roots and Powers of Simple Algebraic Quantities. +2ab8ab2+86 Cube a3+6a2b+12ab2+863 a a2x Square == a1-2a1x+x2 a2-x a—2a1x+ a2x2 Cube-a-3a*x+3a2x2-x3 It is very well known that the value of the figures in the common arithmetical scale increases in a tenfold proportion from the right to the left; a number, therefore, may be expressed by the addition of the units, tens, hundreds, &c. of which it consists. A number of 2 figures may be expressed by 10a++b. .3 figures. ...by 100a10b+c. ..........4 figures............by 1000a + 1006+10c+d EVOLUTION. 55. EVOLUTION is the reverse of Involution, and consists in finding the square, cube, &c. roots of any given quantity. CASE I. To extract the roots of a simple quantity, or powers. RULE. Extract the root of the numerical coefficient, if it have any, as in common arithmetic; then divide the index of the given power by 2, for the square root, 3 for the cube root, 4 for the biquadrate root, and depending on the index of the root required; thus the square root 9x3-3x2-3x=-3x; and the cube root of 8x=2x+2x2. 1 If the coefficient be a fraction, extract the root both of its nu4 2 1 merator and denominator; thus, the squarex=x, or x=x. 3 2 1. Find the square root of 16a2x2=16×a2x2, or 16a2x2/16Xa2X2=+4×a×x=±4ax, Ans. 2. Find the cube root of Sa3x3, or (8Xa3×23). ../8a3x3=8×¥ a3×~/x3=2XaXx=2ax, Ans. 3. Find the cube root of 125a3x, or -125Xa3 Xx®. ..—125a3x=-125a3×2/x®——5×a×x2-5ax2 32x10. Ans. 243a15 7. Find the fourth root of 81a1b, or 256ax. 3abb, or 4ax2. NONINY 3xy -64a3b= 4ab2, or -125x125/ax 5ax or 8a3 8a3 2a 125x125x5x2 256a1x3 Ans. 81ab√(81a*b*Xb2) = 3abb, or 256a⭑x=256a/x=4ax, Ans. -32ab =√(—32a5b3×b)——2abb, Ans. or -32a510 243 CASE II. To extract the square root of a compound quantity. RULE. Arrange the terms according to the power of some letter, as in division. Find the a2+2ab+b2+2ac+2bc+c2(a+b+c root of the first term, and set a2 it in the quotient. Subtract 2a+b)2ab+b2 2ab+b2 two terms for a dividend, and take double (or two times) the root already found for a divisor. Divide the dividend by the divisor, and put the result both in the quotient and divisor. Then proceed as in common arithmetic. 1. 9x-12x+16x2-8x+4(3x2-2x+2 9x4 6x2-2x-12x3+16x2 6x-4x+2|12x2-8x+4 2 6x constant trial divisor. First, we extract the square root of 92*, the first term. This gives 32 for the first term of the root required. This we place on the right hand of the dividend, in the manner of the quotient, as in division. Squaring this term, and subtracting it from the dividend, we get for a remainder -12x+16x-8x+4: we now double 32, and place it as a divisor on the left of this remainder, and dividing by it, 6x2, the first term of the remainder, we get or obtain the quotient-2x, the second term of the root sought, which we annex, with the proper sign, to the double root 622; multiply the whole of this quantity, 6x-2x, by -2x, and subtracting the product from the first remainder, we obtain for a second remainder 12-8x+4; then by doubling 322-2x, the two terms of the root thus found, and dividing 12, the first term of the new remainder, by 6x2, the first term of the double root, we obtain +2 for a quotient, which is the third term of the root sought; and by annexing it to the double root 6x2-4x, and then multiplying the whole of this quotient, 6x2-4x+2, by 2, and subtracting the product from the second remainder, we find 0 for a new remainder, which shows that the root required is 3x2—2x+2. 2. 4x+12x+5x*—2x3+7x2-2x+1(2x3+3x2—x+1 2 3. 9x-12x+10x*—10x3+5x2-2x+1(3x3—2x2+x—1 9x6 6x-2x-12x+10x 1st divis. -12x3 + 4x1 6x-4x+x)6x-10x+5x Constant 623 trial divisor. 2d divisor. 6x*— 4x3+ x2 3d divisor. -6x+4x-2x+1 Dividend. 4x3-20x+a+25x*a*+24x*y*b*_60x*y*a*b*+36by 4x5x*a*)—20x+a+25x+a+ 4. 4x3 -20x1a3 1st divisor. -20x 2d divisor. —10x*a*+6xy*¿*)+24x*y*z*—60x*y*a*b*+36by 5. What is the square root of 1+ x ? 1 + x(1+x—{ x2+1622 + $822+ &c. 1 |