Reduce the following to equivalent fractions with rational denominators: 244. When the denominator is a binomial, containing radicais of the second degree only. 1-√1-x rational denominator. Multiplying both terms by 1+√1-x, Multiply both terms of the fraction by the denominator with the sign between its terms changed. 245. The approximate value of a fraction, whose denominator is irrational, may be most conveniently found by reducing it to an equivalent fraction with a rational denomi nator. Find the approximate values of the following to three decimal places : 246. An Imaginary Quantity is an indicated even root of a negative quantity (Art. 201); as, √-4, or α2. In contradistinction, all other quantities, rational or irrational, are called real quantities. 247. Every imaginary square root can be expressed as the product of a real quantity multiplied by √-1. Thus, 2 √-a2=√a2 × (−1)=√a2 × √−1=a√-1; X √-5 =√5 × (-1)=√5√-1; etc. 248. Let it be required to find the powers of √-1. By Art. 198, - 1 signifies a quantity which, when multiplied by itself, will produce - 1; that is, Therefore, (-1)=(√-1)2× √−1 =(−1) x√--1=-√-1; (−1)=(√−1)o× (√−1)=(−1)×(−1)=1; (√−1)=(√-1)*x √-1 = 1 x√-1=√-1; etc. Thus the first four powers of V-1 are √-1, -1, - √ -1, and 1; and for higher powers these terms recur in the same order. MULTIPLICATION OF IMAGINARY QUANTITIES. 249. The product of two or more imaginary square roots may be found by aid of the principles of Arts. 247 and 248. =√6x(-1) (Art. 248) = -√6, Ans. 2. Multiply V-a2, V--b2, and V-c2. √-ax√-b2×√-c=a√-1xb-1xc√-1 =abc(-1)=-abc√-1, Ans. RULE. Reduce each imaginary quantity to the form of a real quantity multiplied by √ - 1. Form the product of the real quantities, and multiply the result by the required power of √-1. 10. √-1, -9, √-16, and V-25. Expand the following: 11. (2-3)2. 13. (1+√-1) (1−−1). 12. (√-3+2√2)2. 14. (a+b) (a-v-b). 15. (x-x+y √ - y) (x √ - x - у - у). Note. The rule of Art. 241 would have given the same result; hence, that rule applies to the division of all radicals, whether real or imaginary. |