XXIII.

Verify the truth of the following identical expressions:
1. (x1+x) (x}+1+x−4)=x3+x¬3+1.
2. (2x+y ̄1)(2y+x−1) = (2x1y*+x ̄1y ̄1)3.
3. (x*—3y+)(x*+y1)(x*+2y*)(x+3y*)(x*—y*)(x*—2y*)

:

=x3-14x2y+49xy3—36y3.

4. {(1+x)(1−x)*—(1−x)(1+x)*}*—{(1—x)*—1}2

= x2{2(1 − x2)1 — 1}.

5. {xy+2x(xy—x2)*}1+{xy—2x(xy—x2)1 } = (2x2—xy)'.
6. {2x2+2(x1—y1)*}*— {2x2—2(x1—y*)*}* = 2(x2—y3)*.
7. (a*+b*—c3)(a1+b*+c1)+(a*—b3 − c¥) (a*—b1+c1) = 2(a+b+c).
8. {(ab)*+(ac)'+(bd)*+(cd)*}.{(ab)* — (ac)* — (bd)*+ (cd)*}

9. (a®+a1b2)1+(bo +b1a2)* = (a2 +b2)§.

10. (a*b+a3bc)*+(ab1+b*c)3 = (a+b) (a+c)*b*.

11. (a2+a3b3)*+(b2+bŝa3)* = (a}+b1)}.

=(a-d)(b-c).

12. {(a+b)}+(a−b)3}(a+b)ŝ—{(a+b)§ — (a—b)§}(a—b)3 = 2(a2 + b2).

XXIV.

Find the values of the following expressions:

1. {(x2+y3)*+z}.{(x2+y)—z}, when x-4, y = 5, z=6.

2. (1−x)*+{1−x+(1+x)'}', when x=

3. x3-2x2+x+18, when x=2+ √−5.

24

25

4. {x1+2x3+3x2+2x+5}', when x2+x+1=0.

5. x+y+(x2—xy+y')', when x = √2+1 and y=√2-1. a+(a2+b2) when a = -4 and b = -3.

a3 — 2b (a2 —b2)'

1+(1+x)*+1+(1−x)i '

x-y x+y, when x=2+ √√3 and y = 2−√3.

6.

7.

8.

x+y+x-y'

x2+xy+y2

√3-1

9.

and y=

x2 — xy+y2'

√3-1

10.

(x2+a2)1 + (x2 — a2)*

when x = (a2+1)'.

(x2+a3)* — (x2 — a2)+

'

+xy+y, when x=
√3+1

11.

{(a3−x3)(a+x)}} — {(a3+x3){(a−x)}3 when x = a.
{(a+x)2(a−x)}} — {(a−x)2(a+x)}}

XXV.

Reduce the following expressions to their simplest forms:

1+x1—x}—x2

1.

2.

x+6x+5x

15. {(a—b)2+4ab}*.{(a+b)2—4ab}2. { "." —b2 — 3ab }

-b

= a2-b2.

17. (1−x2)2 (1+x)2 { (1−x)*+(1+x)' } = (1−x2)(1+x)1. 2(1+x) (1+x) (1-x))

XXVII.

=

1. For what reason is the negative sign necessary for distinguishing algebraical from arithmetical calculation? and how does the algebraical representation of impossible quantities result necessarily from this use of the negative sign?

2. Is it always possible to express the sum of the squares of two algebraical expressions in the form of the product of a sum and difference?

3. Shew that in the addition, subtraction, multiplication, and division of quantities of the form a+b√−1, and also in the involution and evolution of such quantities, the results will always be of the form AB-1.

Express

and

2+3-1 a+b√-1
4+5-1 c+dy-1

in this form.

4. Explain why the introduction of imaginary expressions does not vitiate a process of algebraical reasoning.

5. Show that {√−1}" will have four different values according to the forms of the number m, and exhibit the values of (1+√3).

1 im

6. Shew that (-a)× (—b) is always an impossible expression,

1

1

and (—a)îm+î × (—b)TMm is always a possible expression, when m is any integral number.

7. Are the quantities -ax-b, and /-ax -b, rational or irrational?

8. Shew the absurdity of the following reasoning:

( − 1 )' = ( − 1 ): = {(−1)2 } * = (+1)' = +1.

9. Point out the fallacy in the statement

√(−1) × √(−1) = √ {(−1) × (− 1)} = √ +1 = 1.

10. Find the third power of

inferences may

1-3, and of +1±√3; what

be deduced from the results of the operation? Shew

also that of +1-√-3

2

+1 +2

11. Shew that any power of a cube root of unity, is itself a cube root of unity. Exemplify this property in the fourth and fifth powers. 12. Is the following argument legitimate? If ao = 1 and 6o = 1 ;; therefore ao 6o, and consequently a = b.

-=

13. Establish the equalities, in which w is an imaginary cube root of unity,

(y − z)2(y + z −2x)+(≈−x)3 (s+x−2y)+(x−y)2(x+y−2z)

=(y+z-2x)(z+x-2y)(x+y-22)
=(x+wy+w2 %)3+(x+w3y+w2)3.

XXVIII.

Verify the correctness of the following expressions

1. √(—a2).√(—b2) — —ab,

and (-a).(—b2). ~√/ (—c2) = -abc/1.

2. (—a3). ~/(—b3) =+ab, and †/ (—a3). † ( − b3). 3/ (—c3) = — abc. 3. {√(−2√3)}' = -12.

4. (a+b√−1)(a−b✓✓/−1) = (a+21a3b1+b)(a−23aab3+b).
5. (a2+b2√−1)(a3 — b3 √✓/ −1) = (a2+23ab+b2)(a2—23ab+b2).
6. (a+b√−1)(c+d√−1)(c−d√ − 1)(a—b√ − 1)

= (ac+bd)2+(ad—bc)3.

7. (x−√2—√−3)(x−√2+√−3)(x+√2−√3)

(x+√2+√−3)=x+2x+25.

8. (1+1)+(1-1)=0 and 4-1.
9. √(16+30√−1)+√(16−30 √√✅ − 1) = 10.

10. √{a+b√−1}+√{a−b√ −1} = √✓ {2a+2√(a2+b2)}.

11. (a+b√−1)3±(a—b√ − 1)3 = 2a (a2-362), and 26(3a2-b3)-1.

a-b-1

-=a+b√−1.

a3 — (1 — √/ −1)a2b+(1−√−1)ab2+b3 √ −1 = a2 — ab+b2.

a+b√−1

2.

3.

8-1

1+2-1 1+2√-1

1+2-1 1-2-1

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+a3c)(a+a2b+ac) = a2+b2+c2 — ab — ac—be. 4. If a = {(−1+√ −3) and B = {(−1—√—3); shew that (x+y+z)(x+ay+Bz)(x+By+az) = x3+y3+z3—3xyz. 1+-1, then ao+a‘+a2+1 = 0.

5. If a=

√2

6. If a=(1+3), and 8= (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

2. 14/147 and 13/75.

4.50 and√72.

9. 281 and 7/3.

Determine the products and quotients of the following surds :

1. 448 and 112.

2. 4/8 and 36.

108 and √16.

6. and √.

8. 472 and 132.

3. 12/200 and 13150.

4.

5. and T.

7. 98 and 3/5.

9. 1218 and 3/520.

10.

12.

14.

11. √ and V. 13.6 and 18. 15. 3 and 472. 17. 53 and 18.

20 and √5.

and V.
and V.

16. 3/2 and 23.
18.6 and 18.

-

-