INVOLUTION. Involution is the continual multiplication of a quantity into itself, and the products thence arising are called the powers of that quantity> and the quantity itself is called the root. Or it is the method of finding the square, cube, biquadrate, &c. of any given quantity. RULE** Multiply the quantity into itself, till the quantity be taken for a factor as many times as there are units in the index, and the last product will be the power required. Or, Multiply the index of the quantity by the index of the pow er, and the result will be the power required. * Any power of the product of two or more quantities is equal to the same powers of the factors, multiplied together. And any power of a fraction is equal to the same power of the numerator, divided by the same power of the denominator. x2+ax x3+2ax+a2=square x + a x3+2ax2+ a3x + ax2+2a3x+α3 x3+3ax2+3a3x+a3 =cube x + a x*+3ax3+3a2x2+ a3x + ax3+3a2x2+3a3x+a* x2+4ax3+6a2x2+4a3x+a* = 4th power. The third power of x is x2x3, or x®. The fourth power of 2a3b is 2*xa1*b*, or 16a16. The mth power of a” b is a”” b”. The second power of axis ax2, or ax, that is, ax. 1 The nth power of ax" is or ax. mn or And the mth power of a2+x2|3m is a2+x3|3m a2 + 13, namely, the nth power of the cube root of a2+x3. NOTE. All the odd powers, raised from a negative root, are negative, and all the even powers are positive. Thus, the second power of a is—ax-a+a3, by the rule for the signs in multiplication. The third power of —a is +a3×—ɑ——ɑ3. The fifth power of a is +ax-a=—a3, &c. EXAMPLES FOR PRACTICE. 1. Required the cube of 83y3, Ans. -512xy&. 3. Required the 5th power of a—x. Ans. a-5a*x+10a3x2-10a2x3+5axa—x3. SIR ISAAC NEWTON'S Rule For raising a bonomial or residual quantity to any power whatever.* 1. To find the terms without the coefficients. The index of the first, or leading quantity, begins with that of the given power, and decreases continually by 1, in every term to the last; and in the following quantity the indices of the terms are O, 1, 2, 3, 4, &c. 2. To find the unica or coefficients. The first is always 1, and the second is the index of the power; and in general, if the coefficient of any term be multiplied by the index of the leading quantity, and the product be divided by the number of terms to that place, it will give the coefficient of the term next following. Note. The whole number of terms will be one more than the index of the given power; and, when both terms * This rule, expressed in general terms, is as follows: n ガー a+b =a"+n. a"−1b+n. "— 1 an➡262+n. &c. 2 NOTE. The sum of the coefficients, in every power, is equal to the number 2, raised to that power. Thus, 1+1=2, for the first power; 1+2+1=4=22, for the square; 1+3+3+1=8 =23, for the cube, or third power; and so on.. R of the root are +, all the terms of the power will be +; but if the second term be, then all the odd terms will be +, and the even terms — EXAMPLES. 1. Let a+x be involved to the fifth power. and the coefficients will be or 1, 5, 10, 10, 5, 1; And therefore the 5th power is 2. Let x-a be involved to the sixth power. The terms without the coefficients will be x*, x3α, x1ɑ2, x3a3, x2a1, xa3, a ; 6×5 154 20X3 15x2 1, 6, 6x1 59 ; 6 or 1, 6, 15, 20, 15, 6, 1 ; And therefore the 6th power of x-a is 3. Find the 4th power of x-a. Ans. x4x3a+6x3a2-4xa3+a*. 4. Find the 7th power of x+a. Ans. x7+7xa+21x* a3 +35x*a3+35x3a*+21x3a3+ 7xa® +a1. |