266. The quotient of one surd divided by another may be found by rationalizing the divisor; that is, by multiplying the dividend and divisor by a factor that will free the divisor from surds. This method is of great utility when we wish to find the approximate numerical value of the quotient of two simple surds, and is the method required when the divisor is a compound surd. 1. Divide 3√8 by √6. 3√8 6√2 6√2×√6 6√12 = = 6 =√12 = 2√3. 2. Divide 3√5-4√2 by 2√5+3√2. Multiply the dividend and the divisor by 2√5-3√2, (3√5-4√2) (2√5 – 3√2) 54-17√10 (2√5+3√2) (2√5-3√2) 20 18 54-17√10 = 27 - √10. Hence, 2 267. When the divisor is a binomial containing surds of the second order only, Multiply the dividend and the divisor by the divisor, with the sign between the terms changed. Given √2 = 1.41421, √3= 1.73205, √5 = 2.23607; find to four places of decimals the value of: 268. Any power or root of a radical is easily found by 3 (2√a)2 = (2a)2 = 22 a3 = 4 a3 = 4√a2. 2. Find the cube of 2 √a. (2√a) = (2a) = 23 a3 = 8 a3 = 8 ava. 3. Find the square root of 4 x √a3b8. 4 (4 x √a363)* = (4 xa*b*)* = 4 xlab = 4 xi*a*b* = 2√a3b3x2. 4. Find the cube root of 4 x Vab3. (4 x Va3b3)3 = (4 xa3b) 3 = 43 x3 = 48 x*a*b* = √16 a3b3x2. Properties of Quadratic Surds. 269. A quadratic surd is the indicated root of an imperfect square, as √2. 270. THEOREM 1. The product or quotient of two dissimilar quadratic surds will be a quadratic surd. Thus, Thus, √ab × √abc = ab√c; Two dissimilar quadratic surds cannot have all the factors under the radical sign alike. Hence, their product or quotient will contain the first power only of at least one factor, and will therefore be a surd. 271. THEOREM 2. The sum or difference of two dissimilar quadratic surds cannot be a rational number, nor can the sum or difference be expressed as a single surd. For, if √a ± √o could equal a rational number c, we should have, by squaring, that is, a + 2√ab + b = c2; a-b. Now, as the right side of this equation is rational, the left side would be rational; but, by § 270, √ab cannot be rational. Therefore, Va ± √o cannot be rational. In like manner it may be shown that Va± Võ cannot be expressed as a single surd Vo. 272. THEOREM 3. A quadratic surd cannot equal the sum of a rational number and a surd. For, if va could equal c+ √b, we should have, by squaring, and, by transposing, a = c2 + 2 c√b + b, 2c√b=a-b - с2. That is, a surd equal to a rational number, which is impossible. 273. THEOREM 4. If a + √b = x + √y, then a will equal x, and b will equal y. For, by transposing, √b - √y = x-a; and if b were not equal to y, the difference of two unequal surds would be rational, which by § 271 is impossible. ... b = y, and a Х. In like manner, if a √b = x - √y, a will equal x, and 6 will equal y. 275. To Extract the Square Root of a Binomial Surd. 1. Extract the square root of 7 + 4√3. A root may be found by inspection, when the given expression can be written in the form a + 2√6, by finding two numbers that have their sum equal to a and their product equal to b. 75 2. Find by inspection the square root of 75 - 12√21. It is necessary that the coefficient of the surd be 2; therefore, 12√21 must be put in the form 75 - 2√62 × 21 ; Two numbers whose sum is 75 and product 756 are 63 and 12. 75-2756=63-263 × 12+12 = (√63-√12)2. Then, 3. Extract the square root of 11 + 6 √2. 11 + 6√2 = 11 + 2√18. Two numbers whose sum is 11 and product 18 are 9 and 2. |