1 1 =6x5+2x-(3x5+2x) - {2x5+100x (45-)} =30+(15+){10+100-(20-)} 1. If a=4, b=3, c=5, d=6, e=1, f=0, find the numerical values of (2) {a+(b+c)−d} {(a−b)—(c–d)} (a−b+c−d) when a=20, b=10, c=4, d=0. (3) {13a-(b-2c+3d−ƒ)}-{(13a-b−2c)—(3d—ƒ)} (4) a-(b−c)—{b−(a−c)}-{3a− (2b+a−c)}, 7. If a=4, b=3, c=2, d=5, shew that the numerical values are cqual, (1) of (a-b) (a–c) (b-c) and ab (a-b)-ac (a–c)+bc (b−c). (2) of (a+b+d) (a+b−d) (a+d—b) (b+d—a) and 2ab × 2ab. 17. If a quantity be multiplied into itself any number of times, the product is called a POWER of the quantity. Thus a xa, which is denoted by a2; a×a×a, which is denoted by a3; axaxax &c. to m factors, which is denoted by am; are all powers of a. They are called respectively the second, third, &c. mth powers of a, according as the quantity appears twice, three times, &c., m times, as a factor. The quantity a itself is often called the first power of a, and is written a or a1. It may be observed that the second power of a, that is a xa or a2, is frequently called the SQUARE of a; and the third power of a, that is axaxa or a3, the CUBE of a. The figures 2, 3,......m, denoting the number of factors which produce the powers, are called EXPONENTS OF INDICES. The meaning of negative and fractional indices will be shewn hereafter. Ex. II. The following examples will serve to illustrate the above definition. 1. If a=1, b=2, c=3, d=4, e=0, (5) [(a3+b3+c3 +d3) {a+b−(c− a)}+a2b+c2d] × {a23 − (b2 + c2) +ď2}, (6) Shew that when a=4, b=3, c=2, d=5, (1) (a+b+c)3+a3+b3 + c3 = (a + b)3 + (a+c)3 + (b+c)3+Gabc. (2) 6b3={(a+4b)3 — (a+b)3} −3}(a+3b)3 — (a + 2b)3}. 18. The SQUARE ROOT of any proposed quantity is that quantity whose second power or square gives the proposed quantity. Its CUBE Roor is that quantity whose third power or cube gives the proposed quantity; and generally the mth root of any quantity is that quantity whose mth power gives the proposed quantity. The sign or / denotes the square root of the quantity before which it is placed. Thus a or a signifies the square root of a. Similarly ,, denote respectively the cube, fourth......mth roots of the 1, ... quantities before which they are respectively placed. These signs are called radical signs. These roots can also be represented by the fractions 1 1 1 2'3' 4 1 &c. and placed above the quantities as indices. Thus 1 m a a a &c. am respectively represent the square root, cube root, fourth , root, &c. mth root of a. The reason of this will be shewn hereafter. Ex. III. The following examples will serve to illustrate the above definition. 1. If a=36, b=16, c=9, d=1, = √√4x4+ √5×5×9×9+243×2×2×2-3/4x4x9×9 = √√2×2×2×2+ √√5×5×9×9+2 3/4 × 4×4 × 2 × 2 × 2 =2+5×9+2×4×2-3×2×3 =2+45 +16-18-63-18=45. 3. If a 16, b=10, x=5, y=1, 4 −3√√2 × 2 × 2 × 2×3×3×3×3 (a−b) (5√a−b) + √√(a−b) (x+y) =(16-10) (516-10)+ √(16-10) (5+1) =6(20-10)+6=6×10+6=66. 4. If a=16, b=10, x=5, y=1, (a− x)2 — (3b− x2) ± √(a−x) (b+y) =(16-5)2 - (3 × 10−52)± √√(16−5) (10+1) =112-(30-25)=√11 x 11 |