Ex. 3. Divide a—41a3—120 by a3+4a3+5. 3 Ans. a*—4a3 +11a3 – Ex. 4. Divide a3+646 by a*+46*. — 24. Ans. a—4a+1+1667. Ex. 5. Divide xa—xy1+x*y—ya by x*—y‡. Ans. x+y. To involve a Radical Quantity to any power. 230. Let it be required to raise am to the nth power. m 1 1 m 2 m =am. 1 1 1 1 1 1 1 The nth power of am is a xam xam, etc., amm n +; etc., m Hence, to involve a radical quantity to any power, we have the following RULE. Multiply the fractional exponent of the quantity by the exponent of the required power. If the radical has a coefficient, let this be involved separately; then reduce the result to its simplest form. If the quantity is under the radical sign, it is generally most convenient to substitute for this sign the equivalent fractional exponent; but if we choose to retain the radical sign, we must raise the quantity under it to the required power. EXAMPLES. 1. Required the fourth power of fat. 2. Required the cube of √3. Ans. 1fa Ans. √3. 3. Required the square of 3√3. 4. Required the cube of 17/21. 5. Required the fourth power of √õ. 7. Required the fourth power of ab Vab. Ans. 3. Ans. 16a29a2b2. 8. Required the sixth power of (a+b). Ans. a2+2ab+b2. 9. Required the value of √(18) x √(25)". 10. Required the value of V(4ab2) × √(2a2b)*. Ans. . Ans. (2ab)*. To Extract any Root of a Radical Quantity. 231. A root of a quantity is a factor which, multiplied by itself a certain number of times, will produce the given quantity. But we have seen that the nth power of a" is a". There fore the nth root of am is am. Hence we derive the following RULE. Divide the fractional exponent of the quantity by the index of the required root. If the radical has a coefficient, extract its root separately if possible; otherwise introduce it under the radical sign. Then reduce the result to its simplest form. If the quantity is under the radical sign, and we choose to retain the sign, we must, if possible, extract the required root of the quantity under the radical sign; otherwise we must multiply the index of the radical by the index of the required root. Operations on Imaginary Quantities. 232. It has been shown, Art. 195, that an even root of a neg ative quantity is impossible. Thus, √−4, √−9, √—5a are algebraic symbols representing operations which it is impossible to execute; for the square of every quantity, whether posifive or negative, is necessarily positive. Quantities of this nature are called imaginary or impossible quantities. Nevertheless, such expressions do frequently occur, and it is necessary to establish proper rules for operating upon them. 233. The square root of a negative quantity may always be represented by the square root of a positive quantity multiplied by the square root of -1. The factor V-1 is called the imaginary factor, and the other factor is called its coefficient. 234. When several imaginary factors are to be multiplied together, it is best to resolve each of them into two factors, of which one is the square root of a positive quantity, and the other √1. We can then multiply together the coefficients of the imaginary factor by methods already explained. It only remains to deduce a rule for multiplying the imaginary factor into itself; that is, for raising the imaginary factor to a power whose exponent is equal to the number of factors. The first power of √-1 is √-1. The second power, by the definition of square root, is -1. The third power is the product of the first and second powers, or -1x-1=-√-1. The fourth power is the square of the second, or +1. The fifth is the product of the first and fourth; that is, it is the same as the first; the sixth is the same as the second, and so on; so that all the powers of √1 form a repeating cycle of the following terms: : 1. Multiply √9 by √−4. 2. Multiply 1+√-1 by 1-√-1. 3. Multiply √18 by √−2. 4. Multiply 5+2√−3 by 2−√ −3. 5. Multiply a√b by c√—d. 6. Multiply 1-√-1 by itself. Ans. 2. Ans. -ac√bd. Ans. -2V-1. 7. Multiply 2√3-√−5 by 4√3–2√−5. Ans. 14-8-15. 8. Multiply a+√b√−1 by a−√õ√ −1. 9. Multiply av—a2b3 by √—a1b3. 10. Multiply Va+√−b by √—a—√—b. 11. Multiply Ans. a2+b. -17+√-19 by √-119-√-133. Ans. 2√7. 235. Division of Imaginary Quantities.—The quotient of one imaginary term divided by another is easily found by resolv ing both terms into factors, as in the preceding article. Ex. 1. Let it be required to divide √-ab by V-a. Ex. 6. Divide √−12+√−6+√—9 by √−3. Ex. 7. Divide 2√8-√—10 by V-2. Ans. √5+4√=1. To find Multipliers which shall cause Surds to become Rational. 236. 1st. When the surd is a monomial. The quantity Va is rendered rational by multiplying it by √a. For √ax √a=a*xa*=a. So, also, a is rendered rational by multiplying it by a3. 3 Also, a is rendered rational by multiplying it by a*; and an by multiplying it by a1-". 1 Hence we deduce the following RULE. Multiply the surd by the same quantity having such an exponent as, when added to the exponent of the given surd, shall make unity. 237. 2d. When the surd is a binomial. If the binomial contains only the square root, multiply the given binomial by the same terms connected by the opposite sign, and it will give a rational product. Thus the expression √a+√b multiplied by √a-√b gives for a product a-b. Also the expression Vā+ multiplied by Va-√b gives for a product Va-√b, which may be rendered rational by multiplying it by √a+√õ. In general, Va±√√ may be rendered rational by successive multiplications whenever m and n denote any power of 2. When m and n are not powers of 2, the binomial may still be rendered rational by multiplication, but the process becomes more complicated. Ex. 1. Find a multiplier which shall render √5+ √3 rational, and determine the product. |