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√б, by finding two numbers that have their sum equal to a and their product equal to b. 2. Find by inspection the square root of 75 – 12√21. It is necessary that the coefficient of the surd be 2; therefore, 75 — 12√21 must be put in the form 75 – 2√62 × 21; that is, or 75-2√756. Two numbers whose sum is 75 and product 756 are 63 and 12. Then, That is, 75-2√756 = 63 — 2√63 × 12 + 12 = (√63-√12)2. √63-√12 = the square root of 75 — 12 √21 ; 3√7 −2√3 = the square root of 75 — 12√21. 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. Equations Containing Radicals. 276. An equation containing a single radical may be solved by arranging the terms so as to have the radical alone on one side, and then raising both sides to a power corresponding to the order of the radical. 277. If two radicals are involved, two steps may be CHAPTER XVIII. IMAGINARY EXPRESSIONS. 278. An imaginary expression is any expression which involves the indicated even root of a negative number. It will be shown hereafter that any indicated even root of a negative number may be made to assume a form which involves only an indicated square root of a negative number. In considering imaginary expressions we accordingly need consider only expressions which involve the indicated square roots of negative numbers. Imaginary expressions are also called imaginary numbers and complex numbers. In distinction from imaginary numbers all other numbers are called real numbers. 279. Imaginary Square Roots. If a and b are both positive, we have √ab = √a × √b. If one of the two numbers a and b is positive and the other negative, it is assumed that the law still holds true; we have, accordingly : √− 4 = √4 (− 1) = √4 × √=1=2√=1; 1; √ a = √a (−1) = √ax √=1=a*√-1; and so on. It appears, then, that every imaginary square root can be made to assume the form a√1, where a is a real number. 280. The symbol √-1 is called the imaginary unit, and may be defined as an expression the square of which is 1. Hence, 1x V-1 = (√− 1)2 = −1; √-ax√_b = √a × √ 1 × √ō x √− 1 = √a × √bx (√−1)2 = √ab x (-1) 281. It will be useful to form the successive powers of the imaginary unit. 1) 1)2 + √-1; = 1; — 1)3 = (√= 1)2 √= 1 =(−1) √=1=-√−1; √— 1) + = (√— 1)2 (√− 1)2 = (− 1) (− 1) = +1; √— 1)3 = (√— 1) + √=1 = (+1) √=1 = + √−1; and so on. If, therefore, n is zero or a positive integer, 282. Every imaginary expression may be made to assume the form a+b√-1, where a and b are real numbers, and may be integers, fractions, or surds. If b = 0, the expression consists of only the real part a, and is therefore real. If a 0, the expression consists of only the imaginary = part b√1, and is called a pure imaginary. |