39 Letox-a be involved to the six th power. و and the coefficients will be 1, 6, 6M5, 15×4, 2083, 15×2, 6x1 2 3 4 5 5 or 1, 6, 15, 20, 15, 6, 13 And therefore the 6th power of x-a is 2 -6x+15-20x3a2 +15x6xa3+a. 3. Find the 4th power of x-d. 22 Ans. x- 4x3a+6xaxa3+a*. 4. Find the 7th power of x-ta. Ans. x2+7xa+21x5a2-+35x4a3+35x3a4+218*4 +7xa+a7. 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 ta is either +a, or -a; +ax+a=+a, and1-axta2 also. 2 for And 3 EXAMPLES. 2 1. The square root of 9x = 3x2 =3x. 2 To find the square root of a compound quantity. RULE. 2 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 ta' is +a; and the cube root of -a3 is -a; for +ax+x+a=+a3; and -a x-ax-a-a3. 3 Any even root of a negative quantity is impossible; for neither +ax+a, nor -ax-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 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. 2. Extract the square root of 4-4x3+6x2-4x+1 3. Required the square root of a2 + 4a3x+6a2x2+ 4ax2+x4. Ans. a2 a + tofx2x+3 4. Required the square root 4 2 2 2ax+x2. 1 17 6 1625 -, &c. To find the roots of powers in general. RULE. 1. Find the root of the first term, and place it in the quotient. 2. Subtract the power, and bring down the second term for a dividend. 3. Involve the root, last 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, and subtract and divide as before; and so on, till the whole be finished. EXAMPLES. 1. Required the square root of a2-2a3x+3a2x2-2 a-2a3x+3a2x2-2ax+x (a2-ax+x2 2. Extract 2. Extract the cube root of x+6x5-40x3+96x-64, x+6x-40x3+96x-64(x2+2x-4 3x4)6x5 x+6x5+12x+8x3 3x4)-12x4 x+6x5-40x3+96x-64 * 3. Required the square root of a2+2ab+2ac+b2+2ba Ans. a+b+c +c2. 4. Required the cube root of 5-6x+15x-20x3+ 13x2-6x+1. Ans. x2-2x+ 5. Required the biquadrate root of 16a-96a3x+216 x2-216ax3+81x4. 22 Ans. 20-3 SURDS, SURDS are such quantities as have no exact root, being usually expressed by fractional indices, or by means of the radical sign ✓. Thus, 2, or 2, which denotes the square root of 2. 2 And 33, or √3", signifies the cube root of the square of 3; where the numerator shews the power, to which the quantity is to be raised, and the denominator its root. PROBLEM |