2. Find a factor that will make 8-1-3 rational. 1. Find a multiplier that shall make 5-2 rational. Ans. No5+No2. 2. Find a multiplier that shall make 7+6 rational. 3. Find a multiplier that shall make 10-2 rational. Ans. 10+2. 4. Find multipliers that shall make Noa+b+√c rational. Ans. Na-No-√√√c, and (a−b-c+2bc). 5. Find a multiplier that shall make 3-1 rational. Ans. (√3+√1)(√3+√1). PROBLEM XIV. ART. 211. To reduce a fraction, whose denominator is a surd, to another that shall have a rational denominator, without changing its value. RULE 1. When the proposed fraction is a simple one, multiply each of its terms by the denominator. 2. If it be a compound surd, find such a multiplier by the last Art. as will make the denominator rational, then multiply both the numerator and denominator by it. 212. To change a binomial, or residual surd, into a general surd. RULE. Involve the given binomial, or residual, to a power corresponding with that denoted by the surd; then write the radical sign of the same root over it. EXAMPLES. 1. It is required to reduce 2+3 to a general surd. 4. Let 3-5 be reduced to a general surd. Ans. √(14—6No5). 5. Let No2+2√6 be changed to a general surd. 6. It is required to change 4—√√7 to a general surd. Ans. (23—87). 7. Let 7/3-3/9 be changed to a general surd. PROBLEM XVI. TO EXTRACT THE SQUARE ROOT OF A BINOMIAL SURD. 213. A binomial surd is one in which one of the terms, at least, is irrational; as a±√√√, or √√a±√√√b. To extract the square root of a+√, we put Multiplying the two first equations together, We have (a+Noõ)XN (a−√√/b)=(m+n)X(m—n). And (a2-b)=m2-n2. Having both the sum and difference of m2 and n2, we obtain, by addition and subtraction, the following equations: (a2—b) 2 2 Consequently, ~(~+~/b) =~(~+~ (@2 -3)) + ́a—No (a2 —b)` 2 (a2—b) (a2 + And √ (a−√_b) =~ (~+~ (@2 —b)) — ~ (a—~ (a" —b)). It is certain that both a and √(a2—b) must be rational, in order that the expressions within the parentheses may be |