5. Divide 12/15 by 43. Quot. 35. 3 by 14. Quot. 130. 3 10. Divide x by x. Quot. 3 3 11. Divide 16—3 √/9 by 4+3 √3. Quot. 4—3 √3. 69. INVOLUTION OF SURDS. RULE I. Involve the coefficient to the power required, for the rational part of the power, (Art. 265 to 267. Part I.) II. Multiply the index of the surd by the index of the power to which it is to be raised, and the product will be the surd part. III. Annex the rational part of the power to the surd part, and the result will be the power required 2. EXAMPLES. 1. Involve 23x to the fourth power. part. Thus 1=2×2×2×2=16 the rational part. 3 And x'}}1=x'1× *=x3=3√x1=3√x3×x=x3 √x the surd Whence 16xx3 √√x=16x3 √x, the power required. 3 -- 27 the rational part. 3 And √5=5=5=125+=25×5;=5/5 for the surd part. z This rule is founded on the same principles with involution of rational quantities, (Art. 52.) and multiplication of surds, Art. 67. 9 x2 −24 xy √√z+16 y°z the square required. 4. Involve 23√2 and 3/3 x each to the square. Ans. 43/4 and 27 x. 70. EVOLUTION OF SURDS. RULE I. Extract the required root of the rational part for the coefficient. II. Multiply the index of the surd into the (fractional) index of the root to be extracted, for the index of the surd part. III. Annex the rational part to the surd for the root required". EXAMPLES. 1. Extract the square root of 163/9. Thus 164 the rational part, and 94×÷9 the surd part; whence 469 is the answer. 2. Required the fourth root of 81 a3y z. Thus *√81=3, and * √a3y°z=a‡y‡zi=a‡yizi. This rule depends on the same foundation with evolution of rational quantities, Art. 56. 3. Required the square root of 1−2√2+2? Thus 1-2/2+2 (1-2 the root. 4. Extract the square root of 9'/3 and of a". Roots 6. Required the square root of x3-2x√y+y? Root x-√y. EQUATIONS. b 71. An algebraic equation is an expression whereby two quantities (either simple or compound) are declared equal to each other, by means of the sign them. of equality placed between 72. In equations consisting of known and unknown quantities, when the unknown quantity is in the first power only, the expression is called a simple equation; when it is in the second power only, it is named a pure quadratic; when in the third power only, a pure cubic, &c. But when the unknown quantity consists of two or more different powers in the same equation, it is then named an adfected equation. 73. REDUCTION OF SIMPLE EQUATIONS, INVOLVING ONE UNKNOWN QUANTITY ONLY. The business of equations is to find the value of the unknown quantities concerned in the equation, by means of those that are known; this process is called Reduction, and its operations are founded on the following self-evident principles: namely, if equals be added to, subtracted from, multiplied into, or divided by equals, the results will respectively be equal. The word Equation is derived from the Latin æquus, equal. The reduction of an equation consists in managing its terms so that the unknown quantity may, at the end of the process, stand alone and in its first power, on one side of the sign =, and known quantities only on the other: when this is effected, the business is done; for the value of the unknown quantity is found, it being equal to the aggregate of the known quantities incorporated together, according to the import of their signs. 74. To transpose the terms of an them from one side of the sign equation, that is, to remove of equality to the other. RULE. Make a new equation, in which place the quantity to be transposed on the opposite side of the sign, to that on which it stood in the preceding equation, observing to change its sign from + to or from to +; and let the rest of the quantities stand as in the preceding equation. EXAMPLES. 1. Given 4+5-2=6+3+7−9, to transpose the terms. OPERATION. 1st step. To transpose-2. thus 4+5=6+3+7-9+2. 2nd step. To transpose+5....4=6+3+7—9+2−5. 3rd step. To transpose+6. 4-6=3+7-9+2—5, 4th step. To transpose+3. ... 4-6-3 7-9+2-5. 4-6-3-7=-9+2-5. 7th step. To transpose+2. 4-6-3-7+9-2=-5. ... 4-6-3-7+9-2+5=0. This rule is founded on the following self-evident principle; namely, "If equals be added to equals, the sums will be equal;" for transposition is neither more nor less than adding equals to equals: thus in ex. 1. there is given the sum is ....... 4-6=3+7-9+2-5, as in the fourth step. And so on throughout the operation: whence it appears, that transposition is equivalent to adding the quantity to be transposed, with a contrary sign, to both sides of the equation; and consequently that the quantities resulting from this addition are equal. Explanation. In the first step we observe that the-2, being transposed, becomes +2; in the second step +5, being transposed, becomes-5, &c. it must be likewise observed, that transposing does not affect the equality; the quantities on one side the = being equal to those on the other, as well after transposing, as before: thus, in the given equation, the aggregate of the numbers on each side the = is 7; in the first step it is 9, in the second 4, in the third-2, in the fourth -5, in the fifth-12, in the sixth-3, in the seventh-5, and in the eighth 0 nothing; in every step the sum of the numbers on one side the =, incorporated together according to their signs, is equal to that of the numbers so incorporated on the other. 75. When several quantities are to be transposed, it is not necessary to take them one at a time, as in the foregoing operation; they may be transposed all together, observing to change the sign of each of them. 2. Given a+b-c=d, to transpose b—c d. Thus a=d-b+c, the answer. 3. Given x-b-z+y-4, to transpose -z+y-4. Thus x+z-y+4=b, the answer. 4. Given a+b-c-3-y, to transpose +b-c and-y. Thus, to transpose+b-c....a=3—y—b+c. And to transpose -y....a+y=3—b+c. 76. The unknown quantity in an equation being connected with known ones by the sign + or - 2 to find its value. RULE. Transpose all the known quantities which are con nected with the unknown one, and (Art. 75.) collect them together into one, according to their signs; the result will be the value of the unknown quantity". d This operation (as was observed in the preceding note) is equivalent to adding bc to both sides: thus to the given equation a+b―c=d add b+c= −b+c the sum is a = d-b+c, the ans. Again, in ex. 3. to the given equation x = b−2+y−4 and the same of other examples. the sum is x+2−y+4=¿ • In the foregoing rules we are taught how transposition is performed, here we learn its object; namely, to get the unknown quantity by itself on one side of the equation. In the operation of ex. 5. we have x=4-1+2−3; now 4-12-3 equals 6-4, and 6-4 equals 2, therefore r equals 2, because "things that are equal to the same are evidently equal to one another." |