+x, namely, the nth power of the cube root of 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 isaata?, by the rule for the signs in multiplication. The third power of The fourth power is The fifth power of a is a X-α-a3, a3 X-aat. a is a Xa——as, &c. EXAMPLES FOR PRACTICE, 1. Required the cube of -8xy3. Ans. —512xy3 SIR ISAAC NEWTON'S RULE For raising a binomial 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 * This rule, expressed in general terms, is as follows: given decreases continually by i, in very' given power, a es 2. To find the unice 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 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 ax be involved to the fifth power. The terms without the coefficients will be a3, a^x, a3x3, a2x3, ax1, x3 ; and the coefficients will be 1, 5, 5X4, 10X3, 10×2, 2 3 4 or I, 5, 10, 10, 5, 1; And therefore the 5th power is 2. Let 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= 823, for the cube, or third power; and so on. 308 Let xa be involved to the sixth power. The terms without the coefficients will be , 2 and the coefficients will be 1, 6, 6X5, 15x4, 2013, 15X2, 6x1 3 4 5 or 1, 6, 15, 20, 15, 6, 1; And therefore the 6th power of x-a is **→→6x§3â ̃†15×1a2 —➡20x3a3 +15x*a*—5xa3+a*. α 2 3. Find the 4th power of x-a. Ans. 4. 4x3 af 6x3 a2 - 4פa3 +aa• 4. Find the 7th power of x-a. α Ans. x+7x+21x5a2-+35**a3+35x3a*+21**«$ +7xa+a2. 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. Any even root of an affirmative quantity may be either + or thus, the square root of a is either a, ora; for +ax+a+a", and 1-axta2 also. And EXAMPLES. 1. The square root of 9x2=3x2=3x. 2. The cube root of 8x3=2x 3 = 2x. 2 6 2 31 3. The square root of 3a*x*=a*x3- √✅/3=ax3 √✅/3• 4. The cube root of -125a3×6= — 5a3× 4 8 -5ax* 5. The biquadrate root of 16a82a+x+=2ax2 CASE 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 quo tient. 2. Subtract the square of the root, thus found, 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 result in the quotient. 4. Multiply 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 tax+ex+a+a3; and ---a Any even root of a negative quantity is impossible; for neither +axta, nor -ax-a, can produce —a3. 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 denominator. 317 4. Multiply the divisor and quotient by the term last put in the quotient, and subtract the product from the dividend; and so on, as in Arithmetic. EXAMPLES." 1. Extract the square root of 4412a3x+13a2x2+ 6ax3+x+. ̧* ̧4a2+12a3x+13a2x2+6ax3+x1(2a2+3a*+x2 2 2. Extract the square root of 4-4x3 +6x-4x+1. x2-4×3+6x2-4x+1(x2 —2x+1 3 3. Required the square root of a1+4a3x+6a2x2+ x2 Ans. a2+2ax+x2. 4. Required the square root of 2x+3+ Ans. xx 5. Required |