of the unknown quantities in terms of the other unknown, and then substitute this value, instead of it, in the other equation, and then solve this equation for the other unknown quantity; then the remaining unknown quantity will easily be found, either by substituting this value, or otherwise, as may be found most convenient. If the sum, or the sum of the squares of the two quantities be given, and their product be either given or can be found, the sum and difference of the quantities can be found, and then their values determined by the solution of a simple equation. EXAMPLE 1. Given x+y=21, and=2; to find x and y. From the second equation we find x-2y2; substituting this value instead of x in the first it becomes 2y2+y=21. Hence 16y2+8y+1=169, by comp, sq. 4y+1=13, extract the root. These values substituted instead of y in the first equa tion give x=18, or 24. EXAMPLE 2. Given x+y=10, and xy=24. x2+2xy+y2=100. Squaring the first gives Four times the second 4xy=96. Hence by subtraction, x2-2xy + y2=4. By extracting the root, By the first equation, x-y=2(a). ... 2x=12 (a)+(b). .. 2y=8 (b)—(a) and y=4. EXAMPLE 3. There are two numbers, the difference of whose cubes is 117, and the difference of the numbers themselves is 3, what are those numbers? Let x=the greater, and y=the less; then by the ques Divide the first equation by the second, and there results Square of the 2d x2+xy+y2=39 (a). .. 3xy-30, by subtraction. x2+2xy+y2=49 (a)+(c). But x-y-3. Hence 5, and y=2. EXAMPLE 4. What two numbers are those whose sum multiplied by the greater is 77, and whose difference multiplied by the less is 12? Let x the greater, and y=the less. And (x-y)y=xy—y2=12. Assume xvy, then by substituting this value instead of x in each of the equations, they become v2y2+vy2=77. And vy-y-12. The first of these being divided by the second, gives v2y2+vy v2+v 77. Completing the square, 576v2-3120v+4225=529. Extracting the root, 24v-65-23. Either of these values will fulfil the conditions of the 3 question. The first gives a√2, and y=√2, and the second gives x=7, and y=4. EXERCISES. Sx+y=12 to find x and y. Sx-y=7 to find x and y. Ans. x=8, y=4. Ans. x=9, y=2. Ans. x=12, y=5. x2+ y2=169 to find x and y. Ans. x= =9, y=3. ƒ x2-y2=721 to find x and y. 1. Given 2. Given { 3. Given 4. Given 5. Given 6. Given 4x 5 y2 12 (x+4y=63 to find x and y. xy=96 to find x and y. Ans. x=15, y=12. Ans. x=8, y=12. x2-xy=201 to find x and y. xy-y2-16 9. The sum of two numbers multiplied by the greater is 240, and their sum multiplied by the less is 160; what are these numbers ? Ans. 12 and 8. 10. What two numbers are those, of which the sum multiplied by the greater is equal to 220, and the difference multiplied by the less is equal to 18? Ans. 11 and 9, or 10/2 and √2. 11. Find three numbers such that their sum multiplied by the first may be 48, their sum multiplied by the second may be 96, and their sum multiplied by the third may be Ans. 3, 6, 7. 112. 12. Find two numbers such that five times their difference may be equal to four times the less, and the square of the greater, together with four times the square of the less, may be 181. Ans. 9 and 5. 13. What two numbers are those, the sum of whose squares is 34, and the sum of whose fourth powers is 706? Ans. 5 and 3. 14. What fraction is that which is double of its square, and whose numerator increased by one, and the sum multiplied by its numerator, is equal to the denominator diminished by one, and the remainder multiplied by the denominator? Ans.. SURDS. 84. Those roots which cannot be expressed in finite terms, are called surds or irrational quantities. 3 n m Thus the √2, a2, ~/ or 23, a3, 2, are surds, for their values cannot be expressed in finite terms. All surds may be expressed by means of fractional exponents, in which it is evident that the value of the surd will not be altered by multiplying or dividing both numerator and denominator of the fractional exponent by the same number; thus 8—8—(8), or aa for a raised to mr m mr m n mr the rth power, is a n and the rth root of this is a nr; but a quantity raised to the rth power, and then the rth root extracted, is the quantity itself; hence a surd is not altered in value by multiplying both numerator and denominator of its exponent by the same number, and since a has al m mr ready been shown to be equal to a", the surd is not altered in value by dividing both numerator and denominator of its exponent by the same number. The following operations upon surds depend upon this principle. 85. 1st, To reduce a quantity to the form of a given surd. RULE. Raise the quantity to the power denoted by the exponent of the surd, and indicate the extraction of the same root. EXAMPLE. Reduce 2a to the form of the cube root. Here 2a raised to the third power becomes 8a3, and indicating the extraction of the cube root, (8a3), the form required. 1. Express each of the quantities, ax, 3ay, 2a 4a2 5a and y separately in the form of the square root. 4a2 16a4 25a21 (a2x2)*, (9a2y2)*, (ˆa2)3, ('6x), (12), and/ 16c4 За Ans. 9y 16 2. Express each of the quantities, 2a2c,, 4ay2, 3a"x, and separately in the form of the cube root. Ans. 4ac 2x 27a3 (8a%3)3, (274*)*, (64a3y°)3, (27329)3, and (6403)3. 3. Express each of the quantities, -2a, 3an and separately in the form of the fifth y2 64a3c3 13 8x3 3cx ax 243a5n (24,308) 3. NOTE. If a surd have a coefficient, the whole may be expressed in the form of a surd, by raising the coefficient to the power denoted by the surd, and multiplying this power into the surd, then placing the symbol of the root over the whole product. Thus, 3√x=9x, 2(acx)=(8acx). 4. Express 4√ā, 3/ax2, 5(ax2)‡, and 3(x2y3), in the form of simple surds. Ans. √16a, (27ax2)3, (625ax2)4, and (81x2y). 86. 2d, To reduce a surd to its simplest form. RULE. Resolye, if possible, the quantity into two factors, one of which shall be a complete power, the root of which is denoted by the surd; place the root of this factor before the symbol, and it will be the form required. If the surd have a denominator, multiply both numerator and denominator of the fraction by such a quantity as will make the denominator the power denoted by the surd, then extract its root, and place it without the symbol. EXAMPLE. Reduce /27a3 to its simplest form. Here 27a3-9a2x3a, the first factor is a square, extracting its root, and placing it without the symbol, we have 3a3a, the form required. 1. Reduce √32a3, 3/81a7, √125, and (180a31⁄2o), to their simplest forms. Ans. 4a 2a, 3a23/3a, 5/5, and 6ax/5a. 2. Reduce 3/1250x4y5, (96α5x2)‡, and (72x5 y3)‡, to their simplest forms. Ans. 5xy/10xy2, 2a(6ax2), and 6x2y(2xy). 3. Reduce plest forms. ax acx 3 2a2x3 9 Ans. 7a, 3a, /3a. 4. Reduce (322z)3, (5utz2)3, (7a2yo)3, and (5y2:5)3, to 36 ax ay2 8 their simplest forms. Ans. (x2z), (30a), (21ay)3, and (10xy2z)‡. 87. 3d, To reduce surds having different indices to other equivalent ones having a common index. RULE. Reduce the fractional indices to a common denominator, then involve each quantity to the power denoted by the numerator of its fractional exponent, and over the results place for exponent one for a numerator, and the common denominator for a denominator. EXAMPLE. Reduce (2a) and (3a2)3 to equivalent surds having a common index. Since, and 3=%, the quantities are equivalent to (2a)ễ and (3a2), raising each of these quantities to the power denoted by the numerator of its fractional exponent, they become (8a3) and (9a4), which is the form required. 1 1. Reduce (ac) and 5 to equivalent surds having a common index. Ans. (a2c2)s and (125)š. 2. Reduce 43 and 34 to equivalent surds having a com |