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7. axb=a+a+a+etc. to 6 times = 6 times a or ba.
bxQ=b+b+b+etc. to a times = a times b or ab.

- 63 + 25x - 194 - [3y - {4x - (5y - 62)}]
- 63+ 25x – 194 - [3y – {4x - 5y + 63}]
-63 +253 – 194 - [3y - 4x + 5y - 62]

- 65 + 25x – 194 – 3y + 4x - 5y +63

= 29x – 27y. 8. See Mansford, page 14

{a-(6-c+d)} {a-b-(-d)} = {(a-6)+(c-d)} {(a - b)-(c-d)} =(a - b) - (-2)2 (al - 2ab +62) - (2-2cd+d)

ea - 2ab + 62 - + 3cd - aa. If a = 4, b= 3, C= 2, d=1, the last result

= 16 - 24+9 - 4+4-1 = 29 – 29 = 0. 9. 3(a - b)+2(c-a) – 4{6+21c-a)}

= 32 – 35 + 20 - 21 – 46 – 80+ 8a

9a - 76-6c.
(a+b+c): +(6+c-a)+(c+a-6)2
(a+b-c)2 = a + 2ab - 2ac + b2 - abc + c
(6+c-a)2 = 22 - 2ab - 2ac +62 + 2bc + c
(c+a - b)2=a– 2ab + 2ac +62 2bc + c
3a2 - 2ab

2ac + 362 26c + 308 =3(a2 +62 +62) – 2(ab + ac + bc). 10. See Mansford's Algebra, also previous answers in this series. If x= 1, y = -2, 3- 3.

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11. (x + 4)2 + (x - 4)2 + (x + 6)2 + (x - 6)2 – 3(x+8) (x - 8)

(x+4)= 32+ 8x + 16
(x-4)2 = - 8x + 16
(x+6)= x + 12x + 36

2
(x-6)= x - 12x + 36

+104
3(x+8) (x – 8) = 3x2 - 192

+ 296

4.72

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12.

1 – 2x - (1 - {1-(1 - 2x)} - (1 - 2x)] =I - 2x {1-1

-1 + 2x} - 1+2x] = 1 – 2x - (1 - 1+1 – 2x – 1+2x]

= 1 - 2x, and (1 - 2x)% = 1 - 46+ 4x2 13. {(a + b)2 + (a - b)2} {(a+)2 – (a –0)2}

= (a + b)* - (a - b)*
=Q4 + 4a3b + 6a262 + 4ab3 +64 - a+ + 4a3b - 6a2b2 + 4ab3 - 64
= 8a%b+8ab3 = 8ab(a + b).

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GREATEST COMMON MEASURE.

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1.

-a2 as (*
x3 - apx

a-x - 23 Dividing this remainder by a’, we have x - a.

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x2 - a2

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a2
a?

ax

ax

..X-a, G. C. M.

2. G. C. M. of gax?y2 and 15a2xy = 3axy.

** - 4x + 3)37 - 2033 15x+2(3*
+

20x2 (
3.05 – 12x2

-8

- 8) – 8x2 + 6x +2

x2 - 4x

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12).

+ 3x3 + 12x -16(+3
1x4 - 13x2 + 12x

3.48 + 13x2 – 16
3x3 – 39.4 +36
13)13x2 + 39x - 52
mc2 + 3x – 4
- 4) 13x + 12x

12(*-3 3x3 + 3x2 - 4x4

- 3x2
- 3x2

9x + 12 9x + 12

.. G. C. M. = x2 + 3x - 4. If x = 1, both expressions will vanish.

I-13+ 12 = 0, 1+3+ 12 – 16=0.

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5.

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+ 1)204 + a-1

1000+ + 2003 – 4a - 4/5
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2003 5a2 – 4a +1

+
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5a +502 - 0 -1. 5a® + 5a? – a – 1)2018 - 5a - 4a+ 2003 + 2002 – 40 - 4

2( -5)- 25a2 + 5

a

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6. For rule see Mansford, page 44.

- 40%43% – 8x3y – xy2 +248(47 8
4.73

8x2y + 15xy2
8.x2y + 32y3

15y2)5xy2 - 30y8

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2418 + 22x2 +17x + 5)48x8 + 8x2 + 31x+15/2

( 48x3 + 44x2 + 34x +

36x2 – 3x + 5.

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