3. Required the square root of 3a2x6. 4. Required the cube root of -125a3x6. 5. Required the square root of Ans. ax3/3. 9x2y2 4a3 λαγά 6. Required the 4th root of 256a4x3. Ans. 4ax2. 7. It is required to find the 5th root of -32x5y1o. Ans. -2xy2. 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 quotient. 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 both in the quotient and divisor. 4. Multiply the divisor, thus increased, by the term last put in the quotient, and subtract the product from the dividend; and so on, as in common arithmetic. root of -a3 is -a; for (+a)×(+a)×(+a)=+a3, and (—«) X(-a)X(-a): -a3. Any even root of a negative quantity is impossible: for neither (+a)X(+a) nor (—a)X(—a) can produce The nth root of a product is equal to the nth root of each of the factors multiplied together. And the nth root of a fraction is equal to the nth root of the ny. merator, divided by the nth root of the denominator. EXAMPLES. 1. Extract the square root of x4-4x3+6x2-4x+1. x4 —-4x3+6x2-4x+1(x2-2x+1= root. 2. Extract the square root of 4a4+ 12a3x + 13a2x2 +6аx3+x4. 4a4+12a3x+15a2x2+6ax3+x4(2a2+3αx+x2 4a4 4a2+3ax)12a3x+13a2x2 12a3x+9a2x2 4a2+6ax+x2) 4a2x2+6ax3+x4 3. Required the square root of a4 + 4a3x + 6a2x2 +4ax3+x4. Ans. a2+2ax+x2. 4. Required the square root of x42x3+23x2 5. Ans. x2-x+1. It is required to find the square root of a2+x2. X2. 204 366 Ans. a+ &c. 16a5 E CASE III. 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 its power from that term, and bring down the second term for a dividend. 3. Involve the root, last found, to the next lowest 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 is finished. EXAMPLES. 1. Required the square root of a4-2a3x + Sa2x2 -2αx3+x4 a42a3x+3a2x2-2ax3+x4(a2—ax+x2 a4 2a2)-2a3x a4-2a3x+ a2x2 2a2)2a2x2 a4-2a3x+3a2x2-2ax3+x4 As this method, in high powers, is generally thought too labo rious, it may not be improper to observe, that the roots of compound quantities may sometimes be easily discovered thus: 2. Extract the cube root of x+6x5-40x3+96x -64. x6+6x5-40x3-96x-64(x2+2x-4 Ans. a+b+c. 3. Required the square root of a2+2ab+2ac+b2 +2bc+c2. 4. Required the cube root of x-6x5+15x4-20 x3+15x2-6x+1. Ans. x2-2x+1. 5. Required the biquadrate root of 16a4-96α3x ́ +216a2x2-216ax3+81x4. Ans. 2a-3x. 6. Required the fifth root of 32x5—80x480x3 40x2+10x-1. Ans. 2x-1. 1. Extract the roots of some of the most simple terms, and connect them together by the sign + or, as may be judged most suitable for the purpose. 2 Involve the compound root, thus found, to the proper power, and, if it be the same with the given quantity, it is the root required. 3. But if it be found to differ only in some of the signs, change them from to, or from to, till its power agrees with the given one throughout. Thus, in the fifth example, the root 2a-3x, is the difference of the roots of the first and last terms; and in the 3d example, the root a+b+c is the sum of the roots of the 1st, 4th, and 6th terms. The same may also be observed of the 6th example, where the root is found from the first and last terms. SURDS. Surds are such quantities as have no exact root, being usually expressed by fractional indices, or by means of the radical sign✔ placed before them. Thus, 21, or √2, denotes the square root of 2, and 34 the cube root of the square of 3; where the numerator shows the power to which the quantity is to be raised, and the denominator its root. PROBLEM I. To reduce a rational quantity to the form of a surd. RULE. Raise the quantity to a power equivalent to that denoted by the index of the surd, and over this new quantity place the radical sign, and it will be of the form required. EXAMPLES. 1. It is required to reduce 3 to the form of the square root. First, 3x3=32=9; whence/9 the answer. 2. It is required to reduce 2x2 to the form of the cube root. First, 2x×2x2×2x2=(2x2)3=8xo; whence Vsx® or (8x6) the answer. 3. Reduce 5 to the form of the cube root. Ans. (125). 4. Reduce xy to the form of the square root. |