问题1.将以下每一项表示为p / q形式的有理数，其中p和q是整数，q≠0
(i)2 -3
解决方案：
2-3 = 1/23 = 1/2×2×2 = 1/8 (we know that a-n = 1/an)
(ii)(-4) -2
解决方案：
(-4)-2 = 1/-42 = 1/-4×-4 = 1/16 (we know that a-n = 1/an)
(iii)1 /(3) -2
解决方案：
1/(3)-2 = 32 = 3 × 3 = 9 (we know that 1/a-n = an)
(iv)(1/2) -5
解决方案：
(1/2)-5 = 25 / 15 = 2 × 2 × 2 × 2 × 2 = 32 (we know that a-n = 1/an)
(v)(2/3) -2
解决方案：
(2/3)-2 = 32 / 22 = (3 × 3) / (2 × 2) = 9/4 (we know that a-n = 1/an)
问题2.查找以下各项的值：
(i)3 -1 + 4 -1
解决方案：
= 3-1 + 4-1
= 1/3 + 1/4 (we know that a-n = 1/an)
LCM of 3 and 4 is 12
= (1 × 4 + 1 × 3) / 12
= (4 + 3) / 12
= 7/12
(ii)(3 0 + 4 -1 )×2 2
解决方案：
= (30 + 4-1) × 22
= (1 + 1/4) × 4 (we know that a-n = 1/an, a0 = 1)
LCM of 1 and 4 is 4
= (1 × 4 + 1 × 1) / 4 × 4
= (4 + 1) / 4 × 4
= 5/4 × 4
= 5
(iii)(3 -1 + 4 -1 + 5 -1 ) 0
解决方案：
(3-1 + 4-1 + 5-1)0 = 1 (We know that a0 = 1)
(iv)((1/3) -1 –(1/4) -1 ) -1
解决方案：
= ((1/3)-1 – (1/4)-1)-1
= (31 – 41)-1 (1/a-n = an, a-n = 1/an)
= (3 – 4)-1
= (-1)-1
= 1/-1 = -1
问题3.查找以下各项的值：
(i)(1/2) -1 +(1/3) -1 +(1/4) -1
解决方案：
= (1/2)-1 + (1/3)-1 + (1/4)-1
= 21 + 31 + 41 (1/a-n = an)
= 2 + 3 + 4
= 9
(ii)(1/2) -2 +(1/3) -2 +(1/4) -2
解决方案：
= (1/2)-2 + (1/3)-2 + (1/4)-2
= 22 + 32 + 42 (1/a-n = an)
= 2 × 2 + 3 × 3 + 4 × 4
= 4 + 9 + 16
= 29
(iii)(2 -1 ×4 -1 )÷2 -2
解决方案：
= (2-1 × 4-1) ÷ 2-2
= (1/2 × 1/4) / (1/22) (a-n = 1/an)
= (1/2 × 1/4) × 4/1
= 1/8 × 4/1
4 is the common factor
= 1/2
(iv)(5 -1 ×2 -1 )÷6 -1
解决方案：
= (5-1 × 2-1) ÷ 6-1
= (1/51 × 1/21) / (1/61) (a-n = 1/an)
= (1/5 × 1/2) × 6/1
= 1/10 × 6/1
2 is the common factor
= 3/5
问题4.简化：
(i)(4 -1 ×3 -1 ) 2
解决方案：
= (4-1 × 3-1)2 (a-n = 1/an)
= (1/4 × 1/3)2
= (1/12)2
= (1 × 1 / 12 × 12)
= 1/144
(ii)(5 -1 ÷6 -1 ) 3
解决方案：
= (5-1 ÷ 6-1)3
= (1/5) / (1/6))3 (a-n = 1/an)
= ((1/5) × 6)3
= (6/5)3
= 6 × 6 × 6 / 5 × 5 × 5
= 216/125
(iii)(2 -1 + 3 -1 ) -1
解决方案：
= (2-1 + 3-1)-1
= (1/2 + 1/3)-1 (we know that a-n = 1/an)
LCM of 2 and 3 is 6
= ((1 × 3 + 1 × 2)/6)-1
= (5/6)-1
= 6/5
(iv)(3 -1 ×4 -1 ) -1 ×5 -1
解决方案：
= (3-1 × 4-1)-1 × 5-1
= (1/3 × 1/4)-1 × 1/5 (a-n = 1/an)
= (1/12)-1 × 1/5
=12 × 1/5
= 12/5
问题5：简化：
( i )(3 2 + 2 2 )×(1/2) 3
解决方案：
= (32 + 22) × (1/2)3
= (9 + 4) × 1/8
= 13/8
(ii)(3 2 – 2 2 )×(2/3) -3
解决方案：
= (32 – 22) × (2/3)-3
= (9 – 4) × (3/2)3
= 5 × (27/8)
= 135/8
(iii)((1/3) -3 –(1/2) -3 )÷(1/4) -3
解决方案：
= ((1/3)-3 – (1/2)-3) ÷ (1/4)-3
= (33 – 23) ÷ 43 (1/a-n = an)
= (27 – 8) ÷ 64
= 19/64
(iv)(2 2 + 3 2 – 4 2 )÷(3/2) 2
解决方案：
= (22 + 32 – 42) ÷ (3/2)2
= (4 + 9 – 16) ÷ (9/4)
= (13 – 16) / 9/4
= (-3) × 4/9
3 is the common factor
= -4/3
问题6.应该将5 -1乘以多少才能使乘积等于(-7) -1 ?
解决方案：
Let the number be x
5-1 × x = (-7)-1
1/5 × x = 1/-7
x = (-1/7) / (1/5)
= (-1/7) × (5/1)
= -5/7
It should be multiplied with -5/7
问题7.(1/2) -1应该乘以多少，以便乘积等于(-4/7) -1 ?
解决方案：
Let the number be x
(1/2)-1 × x = (-4/7)-1
1/(1/2) × x = 1/(-4/7)
x = (-7/4) / (2/1)
= (-7/4) × (1/2)
= -7/8
It should be multiplied with -7/8
问题8.(-15) -1应除以多少，以使商等于(-5) -1 ?
解决方案：
Let the number be x
(-15)-1 ÷ x = (-5)-1
1/-15 × 1/x = 1/-5
1/x = (1× – 15) / -5
1/x = 3
x = 1/3