299. Like surds may be combined by the ordinary processes of addition and subtraction, that is, by adding the coefficients of the surd and placing the result as a coefficient of the surd. 300. We now proceed to prove a Theorem of great importance, which may be thus stated. The root of any expression is the same as the product of the roots of the separate factors of the expression, that is We have in fact to shew from the Theory of Indices that 301. We can sometimes reduce an expression in the form of a surd to an equivalent expression with a whole or fractional number as one factor. Thus √(72) = √(36 × 2)= √(36).√√/2=6√√2, (128)=(64 × 2)=(64). /2=43/2, (ax)= Na". /x=a.x. EXAMPLES.-CIX. Reduce to equivalent expressions with a whole or fractional number as one factor: 302. An expression containing two factors, one a surd, the other a whole or fractional number, as 3/2, ax, may be transformed into a complete surd. І 303. Surds may be compared by transforming them into surds of the same order. Thus if it be required to determine whether 2 be greater or less than 3/3, we proceed thus: 304. The following are examples in the application of the rules of Addition, Subtraction, Multiplication and Division to Surds of the same order. 1. Find the sum of 18, √/128 and √32. √(18)+√(128)+√(32)=√(9 x 2) +√√(64 × 2) + √√(16 × 2) = 3√2+8√2+4√2 =15√√/2. 2. From 3√(75) take 4√(12). 3 √(75) −4√√(12) = 3 √(25 × 3) − 4 √/(4 × 3) 10. 73/(81)-33/(1029). 20. √(x2+x3y)÷ √(x+2x2y+x3y2). 305. We now proceed to treat of the Multiplication of Compound Surds, an operation which will be frequently required in a later part of the subject. The Student must bear in mind the two following Rules: Rule I. Vax√b=√(ab), Rule II. Vax√α=α, which will be true for all values of a and b. 6. 7√(x+1) by 8√(x+1). 7. 10 by 9√√(x−1). 15. -4√(a2-1) by -2√(a2-1). 8. √(3x) by √(4x). 16. 2√(a2-2a+3) by −3√(a2 −2a+3). 306. The following Examples will illustrate the way of proceeding in forming the products of Compound Surds. Ex. (1) To multiply +3 by √x+2. √x+3 x+3√x +2√x+6 x+5√x+6 Ex. (2) To multiply 4√x+3√y by 4√x-3 √y. 4√x+3√y 4√x-3√y 16x+12√(xy) 16x-9y Ex. (3) To form the square of √(x-−7) −√x. √(x-7)-√x √(x-7)-√x x-7-√(x2-7x) −√(x2−7x)+x 2x-7-2√(x2-7x). |
