XXXIII. Find the second, third, fourth, and fifth powers of the following quantities: 3. 12. Extract the square roots and the cube roots of the following expressions 2, 3, 55, 5/27, 3/2, 3/3, 25/125; and the square roots of 12+6√3, 2+√3, 35-12/6, -5+12√√−1. XXXV. Reduce to their simplest form the following surd quantities:1. 3/147-375-3√3. 3 2.8√√12+4√27-2√1. 3. (15+19/2-−2√3—12√6)+(3+√2+2√3). 4. (√2+√3)(√3+√5)(√5+√2). 5. (9+23+2√5+2√15)*. 6. √(2+3)-√(2−√3). 7. {3√3+2√6}1—{3√3—2√6}2. 8. 73/54+9/250+ 3/2+23/128. 9. 3/192-3/81—3/16+ 3/128. 10. 3/24+/81 + 3/4. 11. (√5+2)}+(√5—2)}. 1. Arrange in order of magnitude 2, 3, 4, 5, 62, without extract ing the roots. √2. 2 2/3 2. Find the values of √3' √2 (13)' ()', (13)', 8-1, 7, (-037), each to four places of decimals. 3. Which is the greater in each pair of the following expressions? /4 or 5 2 or 3: 3/2 or 2√3: (52)} or (53)ł or 5a1: √7 or 24/3: √2 3/5 4. Which is the greater (1) 10+√7 or √19+√3; (2) √2+√7 or 3+√5; (3) √5+14 or 3+3√2? (4) 6-5 or 8-7? (5) 2+5 or 3? 5. Can the three lines whose lengths are 3√3, 5√5, 7√✓7 be tho sides of the triangle? 6. If a denote the length of the edge of each of the five regular solids, shew that (1) the surface of the tetrahedron is a3/3: (2) of the hexahedron or cube is 6a2 (3) of the octahedron 2a√3: (4) of the dodecahedron 15a2√{1+√5}: and of the icosahedron 5a3√3. And (1) the volume of the tetrahedron is .a3: (2) of the hexahe √2 12 dron or cube, a3: (3) of the octahedron 2.: (4) of the dodecahe 3 1. If (a2+b2)+a=bx, find x-x-1 and x+x-1 in terms of a and b. 2. m=x+x ̄1, n=y+y1; then mn+ √(m2—4) (n2 —4) = 2(xy+x ̄1y ̄1). 3. If c-a/1-b2+b/la3, = then shall (a+b+c)(a+b−c)(a+c—b)(b+c—a) = 4a2b2c3. 4. If (ay3 — a2)1 = yz, and (ax2—a2)* = xy, then shall (az2—a2)1 = xz. 5. If x(a2 — y2)*+y(a2—≈2)1 = a2, then x2+y2 = a2. 6. If x1+y1= a3, then (x+y+a)2 = 2(x2+ y2+co2). 7. If x1+y1+≈1a = a1, then 64axy={2(a2+x2+y2+z2)-(a + x + y + z)2}?. 8. Shew that {a+b+c+d'}1 can be found in the form x+y1+z1, when the condition 2a(bed) be+bd+cd is satisfied. 9. If x={r+(y2+q3) d}1+{r—(r2+q3)*}} ; then shall 3+3qx−2r=0. 10. Shew that ax2+by2+cz2 = (a+b+c1)3, when ax3 = by3 = cz3, and x1+y1+% ̄1 = 1. 11. If x+y}+z = 0, then shall (x+y+z)3=27xyz. 12. If x++y++2=0, then (x2+y2+≈23-2xy - 2xz-2yz)3 =128.xyz(x+y+*). 2. Express with negative indices * /a2b3 + a(√√/b)+5/(a-210) + √ a ̃3b•. -e 3. Ii ß= {1=0} 1-6 then shall 1+ß ̄1+√(1 − e2) 4. If a(b—c)2—c(b+c)2= 0, then x) (b x Sa = (a-x)(b-x) A. when x2= ab. _ 6. If (a2+1)(b2+1) _ (+1)(+1); then shall (ab+1)2 7. Shew that = (cd+1)2 ab+1 a-b cannot be equal to ±1, unless a=x and by; a, b, x, y being positive quantities. 8. Shew that the two expressions x-y and √1—x2-√1—y2 fulfil the condition, that the difference of their squares is divisible by the sum of their squares. 9. If yz+xz+xy = 1, prove that { (1 + y2 ) ( 1 + ")" } ' + y { (1 + 3°)(1, +2°) 1+x2 1+ y2 10. If x2+y2+z2+2xyz=1; prove that 1+ (1) {(1—y2)(1—x2) }* + { ( 1 − ≈2)(1 − x2)}1+{(1 − x2)(1 − y2) }1 1. Show that it is not an arbitrary assumption to express the nth 1 root of a by the symbol a, if a" be assumed to denote the nth power of a. 2. If a denote any quantity numerically greater than b, does it follow that (±a)" is always algebraically greater than (±b)"? If not, specify the exception or exceptions. 3. Shew that if any number a be greater than x, a+; than, and a+ is less than the square root of a2+x. 2a+1 4. Is (x+1)" always greater than x*+1? 5. Which is the greater (x+a)(x2+b2)3, or (x+b)(x2+a2)1, supposing a, b, x to be integral positive quantities? 6. Shew that al+ab is greater or less than ab+b, according as a is greater or less than b. 7. If a(1—b3)*+6(1-a) be less than unity, then a2+b2 shall be greater than unity. 8. If a, b, c, be unequal numbers, prove that a+b+c is greater than (ab)*+(ac)*+(bc)*. 9. If x=a+b, and y=c+d, which is the greater (xy) or (ac)*+(bd)*? 10. If x, y, z be positive quantities, any two of which are together greater than the third, then any two of the three quantities {2(y2+x)—x2}, {2(x2+x2)—y2}, {2(x2+y2)—x2}}, are together greater than the third. 11. If (a+√b)*=x‚1+y‚1, and (c+√d)3=x ̧3+y¿1; shew that 12. Shew that 1 is less than unless x n+b 2' 13. If a be greater than b, and b be greater than c, y a-b is never less than a1+b1+c2. XLII. m+n 3. Shew that (x2TM +x2) {a 1.1 1 =xm'n "(xm¬n+xn−m)mn, and x2-1>2n(x”+1—x"−1). X -x 2m-n n+m 772 4. Shew that spectively possible, without remainders. 6. Show that a"-b" and am-b have a common divisor, when m and n denote integral numbers. 7. Write the middle term of the quotients in the division of -bm, m and n being one or both even numbers. 8. If (10,000) = 10, then x = 4, and if 3x=9*, then x=3−. |