3. Required the 5th power of a—x. Ans. a--5x+10a3x2-10a3x3 +5ax1-x3. SIR ISAAC NEWTON'S RULE For raising a bonomial or residual quantity to any power whatever.* The index 1. To find the terms without the coefficients. 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 0, 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 a+b =a”+n.a”~1b+n. "— 1 an—262+n. 2 -I an-363, 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. 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 2. Let x-a be involved to the sixth power. The terms without the coefficients will be x®, xa, x^α2, x3a3, x3a*, xa3, a® ; And therefore the 6th power of x-a is 3. Find the 4th power of x-a. 4. Find the 7th power Ans. x•—4x3a+6x3a3—4xa3+aa. of x+a. Ans. x7+7x6a+21x3a3+35xˆa3+35x3a*+21x3a3+ 7xa® +a1. EVOLUTION. EVOLUTION is the reverse of Involution, and teaches to find the roots of any given powers. CASE I. To find the roots of simple quantities. RULE.* Extract the root of the coefficient for the numerical part, and divide the indices of the letters by the index of the power, and it will give the root required. EXAMPLES. 1. The square root of 9x2=3x3-3x. 2. The cube root of 8x3—2x3—2x. 3. The square root of 3a2x-ax√3-ax3√/3. * Any even root of an affirmative quantity may be either + or : thus, the square root of +a2 is either +a, or -a; for a x+a=+a2, and a×a+a2 also. = And an odd root of any quantity will have the same sign as the quantity itself: thus, the cube root of +a3 is +a; and the cube root of —a3 is -a; for +ax+a×+a=+a3; and -ax - Any even root of a negative quantity is impossible; for neither +ax+a, nora X-a, can produce-a2. Any root of a product is equal to the product of the like roots of all the factors. And any root of a fraction is equal to the like root of the numerator, divided by the like root of the denomina tor. GASE II. To find the square root of a compound quantity. RULE. 1. Range the quantities according to the dimensions of some letter, and set the root of the first term in the place of a quotient, for the first term of the root required. 2. Subtract the square of this root from the first term, and bring down the two next terms to the remainder for a dividend. 3. Divide the dividend by double the root, and set the quotient for the next term of the root. 4. Multiply the divisor and the last term of the root by that term, and subtract the product from the dividend; and so on, as in Arithmetic. EXAMPLES. 1. Extract the square root of 4a*+12a3x+13a2x2+6ax♣ 4a+12a3x+13a2x2+6ax3+x^(2a3+3ax+x2 2x3—4x+1)2x3—4x+1 2x2-4x+1 x“. 3. Required the square root of a⭑+4a3x+6a3x3+4ax3+ . Ans. a3 +2ax+x3. 4. Required the square root of x*—2x3+ + 2 16 1. Find the root of the first term, and set it in the place of a quotient. 2. Subtract the power, and bring down the second term for a dividend. 3. Involve the root, already found, to the next inferior power, and multiply it by the index of the given power for a divisor. 4. Divide the dividend by the divisor, and the quotient will be the next term of the root. 5. Involve the whole root to the given power, and subtract it from the given quantity; then bring down the next term, and proceed as before; and so on, till all the terms of the root be found. EXAMPLES. 1. Required the square root of a*—2a3x+3a3x3-2ax3 +x1. |