CASE V. Subtraction of Radicals. 272.-Ex. 1. From 1/75 take 1/12. SOLUTION. Reducing the given radicals to their simplest form, we have 5√3 and 2/3. Then, 2 times √3 taken from 5 times √3 leaves 3 times √/3, or 3√3. 2. From 125a*b* take b1 8ab* √75=5√3 √12=2√3 3√3 Ans. Saby/b. 273. Rule for Subtraction of Radicals.-Reduce the radicals, if necessary, to their simplest forms. If the radicals are then similar, subtract the coefficient of the subtrahend from the coefficient of the minuend, and annex to the difference the common radical; but if they are not similar, indicate their difference by the proper sign. CASE VI. Multiplication of Radicals. 274.-Ex. 1. Multiply 5ay/8 by 21/6b. SOLUTION. Multiplying the coefficients 5a and 2 together, we obtain 10a; multiplying the radical parts √/8 and √66 together, we obtain √/48b; and we have as the entire product 10a√48b, which, reduced to its simplest form, is 40a3b. 2. Multiply 1/8 by 1/16. SOLUTION. Reducing to radicals 5a√/8×2√6b= == 10a√ 48b=10a√ 16×3b =40a√36 275. Rule for Multiplication of Radicals.-Reduce the radical parts, if necessary, to those of the same degree; then multiply the coefficients together for the coefficient of the product, and the parts under the radical signs for the radical part. If polynomials contain radicals, the process is the same as in multiplication of polynomials. 276.-Ex. 1. Divide 81/108 by 21/6. SOLUTION. Dividing the coefficient of the dividend by the coefficient of the divisor, we obtain 4; dividing the radical part of the dividend by the radical part of the divisor, we obtain √18; and have as the entire quotient 4√/18, which, reduced to its simplest form, is 12√2. SOLUTION. Reducing the given radicals to radicals of the same degree, we obtain 123 and r 277. Rule for Division of Radicals.-Reduce the radical parts, if necessary, to those of the same degree, divide the coefficient of the dividend by the coefficient of the divisor for the coefficient of the quotient, and divide the radical of the dividend by the radical of the divisor for the radical part of the quotient. 278.-Ex. 1. Involve 2a/36 to the second power. SOLUTION. By the definition of involution, we take the given radical twice as a factor, and obtain 4a2√966, which, reduced, is 12a2b3. (2α √ 3b3)2 = 2a√3b3 × 2a√/3b3 ÷ 4a2√9b6 = 4a2 × 3b3 = 12a2b3 2. Involve 5ax to the third power. Ans. 125a3x. 279. Rule for Involution of Radicals.-Involve the rational and radical parts to the required power, and reduce the result to its simplest form. When the quantity to be involved is affected by a fractional exponent, the involution may be performed by multiplying the exponent of each letter by the exponent of the required power. Dividing the index of the root produces the same effect as multiplying the fractional exponent. 1 Ans. 162ab2a3b. 6. What is the fourth power of 1⁄2√4(a2 — x2) ? Ans. a-2a2x2+x1. 7. What is the sixth power of (a+b)3? CASE IX. Evolution of Radicals. 280.-Ex. 1. Find the square root of 4a2/966. SOLUTION. Since the root of a quantity √ 4a2√9b6= ±2a√3b3 is equal to the product of the roots of its factors (Art. 260), we take the square root of the coefficient, which is 2a, and the square root of the quantity under the sign, which is √363; uniting these, we have, for the entire root, ± 2a√3b3. |