from the fact they express respectively the area of a square whose side is a, and the volume of cube whose edge is a. 17. The square root of a quantity is that quantity which, when raised to the second power or squared, will give the original quantity. = It is generally written. Thus, √16 4, 144 = 12. The cube root of a quantity is that quantity which, when raised to the third power or cubed, will give the original quantity. = 3, It is generally written 3. Thus, 38 = 2, 27 √1728 = 12. And so the fourth, fifth, &c., roots are indicated by the symbols,, &c., respectively. 18. The dimensions of an algebraical quantity are the sum of the indices or exponents of the literal factors. Thus, the dimensions of 3 a2b3c1 the dimensions of abx2 = 2 + 3 + 4 = 9, and 1 + 1 + 2 = 4. 19. A homogeneous expression is one in which the dimensions of every term are the same. Thus, a3 + 3 a2b + 3 ab2 + b3 is homogeneous, 2. a3 + 3 a2b+ 3 ab2 + b3; a3 + b3 + c3 3 abc. 3. (3a + 7b) (4 a − 9 b); (a2 + b2) (a + b) (a - b). 4. 6 {2 a3 - 4 ac2 + b2c + bc2. (263 - 2 c3 - ď3)}; a2b + ab2 + a2c + -- 5. (a + b + c + d + e)2; (a2 + 2 ab + b2 — c2) ÷ (a + b+c). a3 + a2d + ad2 + d3 6. ; 3/c3 + 3 c2 + 3 c + 1° 9. (3 x − √x2 + y2)2 (2 x + √x2 + y2 + 2). 10. {5x2 + 2 (y + 2)2} {5 x2 - 2 (y + 2)2}. 11. x + y2 + at 12. (213 – y3) ÷ 3) {3x23 + 3 (3 x2 + 3 xy + y2) y}. CHAPTER II. ADDITION SUBTRACTION, MULTIPLICATION, AND DIVISION. Addition. 20. RULE. Arrange the terms of the given quantities so that like quantities may be under each other; add separately the positive and negative coefficients of each column; take the difference and prefix the sign of the greater, and annex the common letter. (When the coefficients are all positive or all negative, we, of course, simply add them together and prefix the common sign for the coefficient of the sum.) Ex. 3. Add together 5x-3y+ 3y, 6 y2+ 7 xy - 4x, 4 xy + 6 x 5, 2 x2 3 xy + 2. Arranging like quantities in each expression under each other, we have :— Ex. III. Add together 1. 3 a 2b, 4a7b, 2a+3b, a 5 b. - 2. 9 a2+7b3, - 3 a2 + 4 b2, a2 + b2, 4 a2 - 12 b2. 3. a + b + c, 3 a + 2b+3c, 4a+7b a2 6 a 3 a2 xy2 xyz 56 12 ab 2x + 7, x4 2, a2 + 3 a 3 ab 3 b2 + 5a + 106 15. xyz + xx2, x2y xy2 — xyz + x2y2 - xy3, y1 + xy3 + x2y2. 2 a2c 9. x2 + x2y2 + x3y, - x3y 10. a3 + ab2 + ac2 + 2 a2b 2 ab2 2 abc 2 abc 2 bc2. 11. xa − xу3 + xz3 – 3 x3y + 3x3z, 3x2y2 + 3 x2z2 + 3 xy2z + + + - 3 xyz2-6x2yz, y^ — x3y — yz3 + 3 x2y2 - 3x2yz, - 3 xy3 - 3 xyz2 - 3 y3z + 3y2x2 + 6 xy2z, z + x3z - y3z -3x2yz + 3 x2x2, 3 xy2z + 3 xz3 + 3y2x2 - 3 yz3 – 6 xyz2. 21. We have seen, Art. 7 (4.), that the subtraction of a quantity is equivalent to the addition of the same quantity with its sign of affection reversed. We therefore have the follow ing rule: RULE. Change the sign of each term of the subtrahend, and proceed as in addition. 3. Take 5 a2 + 3 ab + 4b2 + 3a +7b+8 from 6a2 + 3 b2 - 2 a. 4. Take 6 a + 8 a2x2 + x1 from 8a+ 6a2x2 + 2 x1. 5. Subtract the sum of the quantities a+ 2 ab2 + b*, a1 — 2 a2b2 + ba from 6 a2 + 8 a2b2 + 6 b4. 6. From 3 + y3 + ≈3 — 3 xyz take 4x3 + y3 + 4 ≈3 + 3 x2≈ + 3 xz2 3 xyz. 3 7. From 3x2 + 3 ax3 9 a2x2 + a3x — aa take 2 x1 + 4 ax3 + 4a3x + aa. 10. Take x1 4x3y + 6 x2y2 4 xy3 + y from x2 + 4x3y + 6 x2y2 + 4 xy3 + y, and subtract the result from their sum. 11. Add together the given quantities in the last example, and subtract the result from 3x2 + 10 x2y2 + 3y2. 12. Take a2 + b2 + c2 + 2 ab + 2 ac + 2 bc from 2 a2 + 2 b3 + 4 ab - c2. Brackets-continued. 22. It was shown, in Art. 8, that, when a quantity inclosed in brackets is to be added, we may remove the sign (+) of addition and the brackets without changing the sign of the terms within the brackets. On the contrary, when the quantity in brackets is to be subtracted, or has the sign minus before it, we must change the sign of every term within the brackets on removing the brackets and the sign of subtraction. We shall now see how to simplify expressions involving brackets connected by the signs of addition and subtraction;— Ex. 1. Simplify (3 a 5 b) - (6 b - 2 c) + + 3 c, or adding together the like quantities, = a -C. Ex. 2. Reduce to its simplest form— (When a pair of brackets is inclosed within another pair, it is con venient to remove the inner one first.) Hence the given expression— = α b + c + {b + a − c = α b + c + b + a C Ex. 3. Simplify the expression a + b + c = a + b + c. The line separating the numerator and denominator of a fraction is a species of vinculum, since it serves to show that the whole numerator is to be divided by the whole denominator. Hence, on breaking up the two latter fractions into fractions having one term only in the numerator, we have 23. As it is often necessary to inclose quantities within brackets, we shall now show how this is done. The following rule needs no explanation : RULE.-When a number of terms is inclosed within brackets, if the sign placed before the brackets be +, the terms must be written down with their signs of affection unchanged; but, if the sign placed before the brackets be —, the sign of affection of every term placed within the brackets must be changed. (When the word sign is used in future, the student is to under stand sign of affection, unless otherwise expressed.) |