XXX.

1. If x=- −1+2√✓✓−1, find the numerical value of x-12x.
2. If a+b√−1 = (a−b √ −1)(x+y√ −1), what are x and y?
3. If a=(-1+√−3),

shew that (a+ab+a2c)(a+a2b+ac) = a2+b2+c2 — ab — ac — be. 4. If a=1(-1+/-3) and 8=(-1-√-3); shew that (x+y+z)(x+ay+Bz)(x+By+az) = x3+y3+z3—3xyz. 1+ √-1, then a+a*+e2 + 1 = 0.

5. If a =

√2

6. If a=(1+√ − 3), and 8 = 1(1—✓ −3), express

(1+a):+(1+8); in the simplest form.

XXXI.

:

Find the sums and differences of the following surd numbers :1. 27 and 48.

3.

405.

180 and 5. 3√ and √ To. 7.500 and 108. 9. 281 and 7/3. 11.500 and 108. 13. 12 and 27.

2. 14/147 and 13/75.

4.50 and ✈√72.

5

6. 9 and 4√3.

7

8. 40 and

135.

Determine the products and quotients of the following surds:

1. 448 and 112.

2. 48 and 36.

108 and }√16.

6. √ and √}.

7. 98 and 3/5.

8. 472 and 132.

9. 1218 and 3/520.

11. and V.

13.6 and 18.

15. 3 and 472.

17. 5/3 and 18.

XXXIII.

Find the second, third, fourth, and fifth powers of the following

Extract the square roots and the cube roots of the following expressions 2, √3, 5√5, 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—3√75—3√}.

3

2.8√√12+4√27-2√16.

3. (15+192-2√3−12√6)+(3+√2+2√3). 4. (√2+√3)(√3+√5)(√5+√2).

5. (9+2√3+25+2√15)*.

6. √(2+√3)−√(2−√3).

7. {3√3+26}-{3√3—2√6}. 8. 73/54+93/250+ 3/2+23/128. 9. 3/192-3/81—3/16+ 3/128. 10. 3/24+3/81 + 3/54. 11. (√5+2)}+(√5—2)§.

12. {(10·4)2+(14·31)2}'.

13. (96) × (243)* × (75)* × (3)1⁄2‰.

1

1. Arrange in order of magnitude 2, 3, 4, 5, 61, without extract

8-1, 7, (037), each to four places of decimals.

3. Which is the greater in each pair of the following expressions? 3/4 or 5 2 or 3: 3/2 or 2√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 2a3: (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.a3: (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+y ̃1;

then mn+ √(m2—4) (n2—4) = 2(xy+x ̄'y ̄1).

3. If c=a/1—b2 +b √✅/1—a3,

then shall (a+b+c)(a+b−c)(a+c—b)(b+c—a) = 4a2b3c3. 4. If (ay3-a2)=yz, and (ax2-a2)=xy, then shall (az2-a2)=xz. 5. If x(a2-y)+y(a2-22) a2, then x+y=a2.

6. If x+y=a', then (x+y+a)2 = 2(x2+ y2+c2).

7. If x1+y1+*=a1,

then 64axy2= {2(a2+x2+y2+z2)−(a+x+y+z)?}2.

8. Shew that {a+b1+c1+d1}1 can be found in the form a1+y+**, when the condition 2a(bed)1 = bc+bd+cd is satisfied.

9. If x={r+(r2+q3)1}3+{r−(r2+q3)*}} ;

then shall

+3qx-2r=0.

10. Shew that ax2+by2+cz2 = (a+b+c), when ar3 by3 cz3,

11. If x}+y+z = 0, then shall (x+y+z)3= 27xyz.

12. If x*+y++z*= 0, then (x2+y2+z2-2xy - 2xz—2yz)3

=128.xyz(x+y+*).

=

2. Express with negative indices

√a2b3+a(√b)+5/(a-2b10)+√ a ̃3ba.

e

3. If ß= {

1+ß‍1+√(1 − e2)
√ (1 − 62) •
cat+c
bat-ct

= 1.

6. If (a2+1)(b2+1) _ (c2+1)(d2+1); then shall cd+1 ±(cd). (ab+1)2

7. Shew that

(cd+1)*
1+ax+by
{(1+a2+b2)(1+x2+ y2)}1

=

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.

(1) {(1—y2)(1—x2) } 3 + { ( 1 − ≈2)(1 − x2) }1+{(1 − x2)(1 − y2)}i

= 2.

1. Shew 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 (±6)"? If not, specify the exception or exceptions.

3. Show that if any number a be greater than x, a+; is greater

than, and a+ is less than the square root of a2+x.

2a+1

4. Is (x+1)" always greater than x2+1?

X 2a

5. Which is the greater (x+a)(x2+b2)3, or (x+b)(x2+a2)*, 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—b2)1+6(1-a2) be less than unity, then a+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)2+(bd)1?

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+x2)—x2}, {2(x2+x2)—y3}', {2(x2+y2)—x2}, are together greater than the third.