11. Find the Product of 5√/ax3a. By reducing the surds to the same root, we obtain 5a3 and 3a2, (239). These are to be multiplied together as before. Ans. 15 Vas. But the given roots of the same quantity a, may also be multiplied into each other, by adding together their fractional exponents and . Thus, 5a3a15a 15 Vas, as before. = Either of these two methods may be applied to roots of the same quantity. The first only is applicable to different roots of different quantities. 20. Find the Product of (3+2√5)× (2−√5). In cases of this kind, in which a Surd is connected with another quantity by the sign + or, the Multiplication, or Division, must be performed as on polynomials. 3+2√5 6+ √5-10=√/5—4. Each term of the multiplicand is multiplied by each term of the multiplier, and the partial products are then added together. Observe that 2√5×−√/5=-2√/25=-10. 21. Find the Product of (2+3√2) × (1+5√2). Ans. 13√2+3: 22. Find the Product of (4— √/3)x(2+3√3). 23. Find the Product of (5+2√/6)×(1+2√6). 24. Find the Product of (1—4√/7)×(3—3√/7). Ans. 10/3-1. Ans. 126+29. Ans. 87-157. 25. Find the Quotient of (√20+√/12÷(√5+√3). Ans. 2. RATIONALIZATION OF SURD DIVISORS. (242.) In computing an approximate value of an irrational numerical expression, it is expedient that a surd divisor or denominator be made rational. For example, suppose we wish to compute an approximate value of 2 divided by the square root of 3. 2 √3 If we extract the square root of 3, for a divisor, a regard to accuracy will require that the root be continued to several figures, and hence will arise the inconvenience of dividing by a large number. By multiplying both terms by the denominator, we have 2 21/3 √12 in which the divisor is rational. = √3 √9 3 The value will therefore be found by taking of the square root of 12; and by this method the computation is much simplified. In pursuance of the object at present in view, it is necessary To find a Multiplier of a given Surd which will cause (243.) 1. A monomial Surd will produce a rational quantity by being multiplied into itself with its exponent subtracted from a unit. Thus a multiplied by a1-, or a3×a¤—a, (241.....3). 2. A binomial in which one or both terms contain an irrational square root, will produce a rational quantity by being multiplied into itself with a sign changed. Thus (√3+√2) × (√ 3−√ 2)=3—2=1. The product in this case is readily found on the principle, that the Product of the sum and difference of two quantities is equal to the dif ference of the squares of the two quantities. 3. A trinomial containing irrational square roots will produce a binomial Surd by being multiplied into itself with a sign changed; and this binomial may be rationalized as above. These principles provide for the most useful cases of the subject under consideration.-In applying them to the rationalization of surd denominators, both terms of the given Fraction must be multiplied by the same quantity, (81). INVOLUTION AND EVOLUTION OF SURDS. (244.) The Powers and Roots of irrational quantities are obtained, or indicated, according to the general principles of Involution and Evolution which have been established in the preceding Chapter. We present here however a particular case of the SQUARE ROOT OF BINOMIAL SURDS. (245.) A Numerical Binomial of the form a±√b admits of a square root in a rational and an irrational term, or two irrational terms, whenever a2b is a perfect square. To determine the method to be pursued in this case of evolution, we must find Formulas for the Square Root of a±√,b. The square of the sum of any two quantities, is equal to the sum of their squares + twice their product, (59). The binomial a+b may therefore represent the square of the sum of a rational and an irrational numerical term, or of two irrational terms, in the square root; a representing the sum of the squares of the two terms, which sum will necessarily be rational,-and b representing twice the product of the two terms. In like manner ab may represent the square of the difference of a rational and an irrational numerical term, or of two irrational numerical terms, in the square root, (60). If therefore we take x and y to represent the two terms of the square root of a±√b, we shall have, (1) x2 + y2=a; Multiplying together equations (2) and (3), we have, (58.) Adding together the (4) and (1), and also subtracting the (4) from he (1), and dividing by 2, we shall find, Extracting the square root of each of these equations, Substituting these values of x and y in equations (2) and (3),and interchanging the first and second members, we have, These are the Formulas required. The right hand member of each will contain at most but two irrational terms, when (a2-b) is a perfect square, that is, has an exact square root. EXAMPLE. To find the Square Root of 6+2 5, or 6+√20, (236.) Substituting 6 for a, and √20 for √b, in Formula (A), And since √36-20-√/16=4, the second member reduces to 6+4 +1 2 6 4 =√5+1, the Root required. The root √5+1 may be verified by squaring it. Formula (B) would give the square root of 6 −2√5. |