MULTIPLICATION AND DIVISION OF SURDS. 81. These operations are performed as in rational quantities, by means of the following formulas: If the surds are of different denominations, reduce them to equivalent ones having a common index, and find their product and quotient by the preceding processes. EXAMPLES. 1. Find the product of 38, 2√6, and 4 √√48. Here 38 x 26 × 4√48 3 × 2 × 4 × √(8 × 6 × 48) = = 24√√482 ÷ 3 4' and also a by a. 13 X 39 9 3 28 24 X 3 27 56 = × 33 = 3/3. 39 117 3. Multiply 32-2/3 by 22+ 3, and divide 5 + √3 by 53. by 5+√3, because the product of 53 and 53 is √32 or 5 3, a rational quantity, and the quotient is obtained in the neatest possible form. 4. Multiply 4 12 by 3/2, and also 3/4 by 12. √ Ans. 246 and 36. Ans. 127 and 4 3. 5. Multiply 23 14 by 34, and divide 6/96 by 38. 4 6. Multiply 2a by a3, and divide — a* by a Ans. 2 a2 and a*. 2. 6 5 5 7. Multiply √2 ab3 by √ 8 ab, and divide 5 a/ax by √ bx. 2 8. Multiply together a b, a (x+2a) and b (x2a). Ans. ab2 (x2 — 4a). 9. Multiply together 3 +5, √2 + √3, √2-3 and 3-5. 10. Multiply (a+b)" by (a + b), and divide 4 x √ a by Ans. (a+b) and 7a x3. 4 by 2 + 2 x +4, and divide 1 Ans. x-4x+16 x 16 and x 2 x3 + 3. 12. Multiply a + a*x+ + a2 x + x2 by x+x*y*+y by x*+x*y*+y* y 2x/3-3x y3 √2. Ans. (x2-3x+2)1 and x y* √3+y x* √/2. 15. Divide 16 x by 2x, and a2 -b by a3 — bo. 16 20 -+ = a; therefore the operations of involution and evolution are expressed as in rational quantities, by the multiplication and division. of indices. and 3 (17 √/21)2 = 17a × 21*** = 17a × 21a = 173 × 21 × 214, .. (1721)3 = 103173/21. 2. Find the square root of 103 and the cube root of 3 3 Here and (1C3)* = 10** * = 10* = 10 × 10 − 10/10, 12 3. Find the square of 1⁄2 a3, and the cube root of √2. 4 14 Ans. and√2. 4 1 11~2. √3 and 2 a b3. ནྟེ Ans. √3,√3, and√2 ab3. 5. Find the square root of 16 a 22 and the cube root of 2 6. Find the fourth power of 2+3 and the square root of 9. Extract the cube root of a3 — 3a2b3 +3 ab − b3. Ans. a — 83. An equation is the expression of the equality of two different algebraical quantities, one or both of which contain some power or powers of the unknown quantity. Thus, 3x+4= 13, a x2 + bx = c, ̧ and x3 + a x2= bx + c, are equations. The members or sides of an equation are the two quantities between which the sign of equality is placed; thus in the equation — a = b- 2x, the first side or member is x a, and the second side or member is b-2x. A simple equation is one which contains the first power only of an unknown quantity, as 2x + 5 = 3x A quadratic equation is one which contains the second power, or both the first and second powers of the unknown quantity, as x2 = or x2+10x= 12. An equation is said to be of the first, second, third, or nth degree, according as the highest power of the unknown quantity contained in it is the first, second, third, or nth power of that quantity; thus, +hx = k is an equation of the nth degree, or of n dimensions. A root of an equation is that quantity which, when substituted for the unknown quantity in it, makes the values of the two sides of the equation the same, and thus verifies the equation. A simple equation has one root only; for if the equation 2x + 3 a = 46 can have two roots, as r, and r2, then by substituting these for the unknown quantity x, we have, 2r,+3a4b, 2 r2+3a4b. Subtracting the latter from the former, we get, 2r1 — 2r2 = 0, or r、 — r2 =0, .. r1 =r2; and the simple equation has consequently only one root. An identical equation is one which can be verified by all values of the quantities contained in it; thus (x + y)2 = x2 + 2xy + y2 is an identical equation, for if any values be given to x and y, the equation I will be verified. The solution of an equation is the method of separating the unknown quantity from the other known quantities in the equation, so that the unknown quantity may form one side of the equation, and a known quantity, or a combination of the known quantities, the other side. SIMPLE EQUATIONS. 84. The unknown quantity is usually so involved in the different terms of an equation as to require several operations before the solution can be effected, and the value of the unknown quantity determined. The solution of all equations depends on the following axioms: 1. If equals, or the same be added to equals, the sums are equal. 2. If equals, or the same be taken from equals, the remainders are equal. 3. If equals be multiplied by the same, or by equals, the products are equal. 4. If equals be divided by the same, or by equals, the quotients are equal. 5. If equals be raised to the same power, the powers are equal. 6. If equals have the same root extracted, the roots are equal. The following remarks, which are founded on the preceding axioms, apply to all classes of equations. 85. A quantity may be transposed from one side of an equation to the other by changing its sign from + to or from to +, without destroying the equality. Thus if x+3= 17, then subtracting 3 from both sides, we have, by axiom 2, x = 173 = 14, where the + 3 has been transferred from the one side to the other with its sign changed to Hence also if xa = b+c, then adding a e to both sides, we have (axiom 1), x-a+a- c=b+c+ ac, or x c = b +a; where the a and c have changed sides, and changed their signs. If the signs of all the terms of both sides of an equation be changed, the equality will still remain, for every term has been transposed, and the sides afterwards interchanged. Hence also if a quantity be found in both sides of an equation with the same sign, it may be removed from each side. Thus if x + c = a + c, then will x = a. 86. Every term of each side of an equation may be multiplied or· divided by the same quantity, without affecting the equality. Thus if+7=9, then multiplying every term by 5, we get 35 10. = x+3545, and transposing x = 45 And if 5x+10= 15, then dividing every term by 5, we get x + 2 = 3, and transposing x = 3—2 =1. Let ==b+c, then multiplying every term by a, we get a x=abac, or xa (b+c). And if axb+c, then dividing every term by a, we have b + c x = a 87. An equation may be cleared of fractions by multiplying first by one of the denominators, then by another, and so on, until all the denominators have been taken away; or multiplying every term by the least common multiple of all the denominators. multiplying by 2, 6 x + 4 x + 3 x + 2 x = 360; hence 15 x 360, and, dividing by 15, gives x = 24. But if we find the least common multiple of the denominators 2, 3, 4, and 6, viz., 12, and multiply every term of the equation by it, we get at once 6x+4x+3x+2x=360; hence 15 x 360, and x = =24, as before. 360 Again, if the equation be x 2x- 3 4, then multiplying by 7, we get 7x-(2x-3)= 28, or removing the bracket, and changing the signs of 2x and 3 into and +, we get 7x 2x+3=28, or 5 x 25; hence x = 5. 88. The student should be particularly careful in equations of this kind, where a fraction is to be subtracted, whose numerator consists of two or more terms; because the line which separates the numerator from the denominator is a species of vinculum, and when that is removed by the removal of the denominator, the whole of the numerator must be subtracted, and the sign of each term must be changed in accordance with the principle of subtraction. Hence also if every term of an equation be either multiplied or |
