ab MULTIPLICATION OF ALGEBRAIC FRACTIONS. 52. RULE. Multiply the numerators together for the numerator of the product, and the denominators together for the denominator of the product. 3a EXAMPLE. Multiply by Here we have 3a xo atb =3ab for the numerator, and (a+b)x(ab)=a?-62 for 3ab the denominator; hence the product required is aa_629 NOTE. Cancel factors which appear in both terms. 3a 2a a 1. Multiply by Ans. 4c 3co 22 36 4a 12ab 2. Multiply by Ans. a 63 a-6 3. Multiply by Ans. a-6 a+bo a+6° 2a+b a-6 2a+b 4. Multiply 23–62 by Ans. (a+b)** 6 a2_62 ab 5. Multiply and together. Ans. a+b' a3_639 a b c d 6. Multiply Žēlē and Ans. a a+bo a ax_s together. together. n DIVISION OF ALGEBRAIC FRACTIONS. 53. RULE. Multiply the dividend by the divisor inverted, and the product will be the quotient required. NOTE. The reason of the above rule will appear from the following example:- Let it be proposed to divide @ by á: for write n, then it will be where the value will not be altered by multiplying nd numerator and denominator by d, which gives ENX x; but n is с equal to ő, therefore nx which is the rule. 3a 2x 3a? 1. Divide by Ans. Ans. d' d d a d X 7a ac a-6 by . x2 41-8 4. Divide by Ans. 5 4.x+2 2x+1 107 5. Divide by Ans. 3 9.-3 6. Divide Ans. atb a-b 7. What is the sum and difference of and ? 2 2 Ans. Sum, a; difference, b. (a+r) 8. What is the sum and difference of and 2 2 Ans. Sum, ao +*2; and difference, 2ax. 1 1 9. What is the sum and difference of and - ? x+w 2.3 2w Ans. Sum, difference, x _W2' 10. What is the product and quotient of a** and a2 x2 1 (a+x)" Ans. Product, :; quotient, 11. Reduce to their simplest forms the following ex 1 1 at 1+ab pressions:-- 1st, +1; 3d, 2x+1 1-a 1+ab x+y 23 23 y 4th, y x+y b+1 y Ans. Ist, =b; 2d, =x; 3d, , x+y x2 wai a-2 a-2 a a 2d, 2. ba INVOLUTION AND EVOLUTION. 54. INVOLUTION is the raising of powers, and EvoluTION the extraction of roots. Involution is performed by the continued multiplication of the quantity into itself, till the number of factors amount to the number of units in the index of the power to which the quantity is to be involved; thus, the square of a is axarao; the 5th power of is xxxxx=x5; the 5th power of (2a) is 2x2x2x2x2 x aaaaa=32a5. 4th powers, TABLE OF THE POWERS OF NUMBERS. 7 9 Squares, 1 4 9 16 25 36 49 64 81 Cubes, 8 27 64 125 216 343 512 729 116 81 256 625 1296 2401 4096 6561 5th powers, 132 243 | 1024 3125 | 7776 1680732768 59049 55. In raising simple algebraic quantities to any power, observe the following rules:-1st , Raise the numerical coefficient to the given power for the coefficient. 2d, Multiplythe exponents of each of the letters by the power to which the quantity is to be raised. 3d, If the sign of the given quantity be plus, the signs of all the powers will be plus; but if the sign of the given quantity be minus, all the even powers will be plus, and the odd powers will be minus. These rules are exemplified in the following Table, which the pupil should fill up for himself : 893 1 + x4 + 1624 a5 -35 32.2.5 as al 8a3 Cubes . . -23 8.23 2763 ao 16a4 4th powers 74 16x4 8164 a5 as 32a5 1 5th powers 75 24365 32a5 1. Raise 2ax?ył to the fourth power. Ans. 16a438y. 2. Raise 7a2px3 to the third power. Ans. 343aRp5xo. 32 3. Raise gabcx to the fifth power. Ans. Fab5clox". closs. 243 4. Raise dažbic to the eighth power. Ans. a472c8. a2bc 2401a864c4 5. Raise 7 to the fourth power. Ans. xyz x+y12 56. When it is required to raise a compound quantity to any power, it can always be effected by multiplying the quantity successively by itself; but there is another means of finding the powers of a binomial, commonly called the binomial theorem, first given in all its generality by Sir Isaac Newton, by which the power required can be written at once, without going through all the intermediate steps. 256 a(n-1) (n-2)(n-3) an *** + This theorem will now be given, and the student can verify its results by actual multiplication in the mean time, as the general proof would not be understood at this stage of the pupil's advancement. BINOMIAL THEOREM. 57. Let it be required to raise (a + x) to the nth power, where n may be any number, and a and x any quantities, either simple or compound. The first term will be d", and the second will be obtained from it, by multiplying by and will therefore be nam-2. The third will be ob. n-1 C and in 2 the same manner, the 4th, 5th, 6th, 7th, will be obtained by multiplying the 3d, 4th, 5th, and 6th successively by n-2 1 n-4 x n-5 I and and so on, so that 3 6 the resulting series will be (a +x)"=a" + na"-23+ n(n-1) an an-$25+ 1:2 1.2.3 1.2.3.4 (x-1)(n-2)(n-3) (n-4) La 50 + &c., where, by substituting 1.2.3 instead of r, 2, 3, 4, 5, successively it becomes (a+x)=a2 + 2ax + x*, From the above, it is evident, (1st), That the power of a in the first term is the same as the power to which' the binomial is raised. (2d), That the powers of decrease by unity in each successive term, whereas those of ac increase by unity, till the last term, where it is equal to the power to which the binomial is raised. (3d), That the number of terms in the expansion is always one more than the power of the binomial. (4th), That the co-efficient of the second term is always equal to the power to which the binomial is raised, and that the successive co-efficients can be obtained by multiplying the co-efficient of the previous term into the power of a in that term, and dividing by the number of terms from the beginning of the expansion; thus 10, the co-efficient of the third term in the expansion of (a + x) can be obtained by multiplying 5, the co-efficient of the second term, into 4, the power of a in that term, and dividing by 2, the number of the term from the begin. 4.5 a Thus, ning of the expansion. In the same manner, may all the other co-efficients be obtained. If the sign of the second term of the binomial were minus, since the odd powers of a minus quantity are minus, and the even powers are plus, the terms which contain an odd power of the second term will be minus, (Art. 55), and those which contain even powers of that term will be plus. (a~*)=a6_6a5x + 15a4x2--2013,3 + 15a24_60x} +26. If it were required to expand (atbnc)", it might be effected by first considering (6—c) as one quantity, and then raising it to the power denoted by its index in each term, and separating into single terms. Thus, (a +6—c)3=a+3a-(6-—c)+ 3a(6—c)? +(6—c)3 a3 =@ +3ab2-6abc+3ac" 63_362c +36c2 — ::{a+(6—c)}=a3+63—(3 + 3a26–3a2c+3ab? +3ac- — 362c +36c2-6abc. 1. What is the 4th power of (a−b)? Ans. a1-4a36 + 6a?y? -4ab3 +64, 2. What is the 3d power of (40—2x)? Ans. 64a5-96aox+48axe-8x?. 3. What is the 9th power of Væ+y? Ans. 23 + 3x2y+3xy? + 43. 4. What is the 5th power of (2a—x)? Ans. 32a5—80a4x+80a3x2 — 40a_x3 + 10ax4-25. 5. What is the 3d power of {a—(x+y)}? Ans. as - 3 - 43-3a-x-3a2y+3ax. +3aye_3x® y —Зху° + баху. EVOLUTION. 58. CASE I. When the given quantity is simple. RULE. Extract the given root of the numerical coefficient for the coefficient of the root, then divide the exponents of each of the literal factors by the name of the root, and the several results connected by the sign of multiplication will be the root sought. EXAMPLE. Extract the 4th root of 81a4x6. The 4th root of 81 is 3, the 4th root of a4 is a'=a, and the 4th root of xo is x1=x . :.3xaxx} =3ax? is the |