问题1.以指数形式写以下每一项：
(i)(3/2) -1 ×(3/2) -1 ×(3/2) -1 ×(3/2) -1
解决方案：
= (3/2)-1 × (3/2)-1 × (3/2)-1 × (3/2)-1
= (3/2)-4 (an × am = an + m)
(ii)(2/5) -2 ×(2/5) -2 ×(2/5) -2
解决方案：
= (2/5)-2 × (2/5)-2 × (2/5)-2
= (2/5)-6 (an × am = an + m)
问题2：评估：
(i)5 -2
解决方案：
= 5-2
= 1/52 (a-n = 1/an)
= 1/25
(ii)(-3) -2
解决方案：
= (-3)-2
= (1/-3)2 (a-n = 1/an)
= 1/9
(iii)(1/3) -4
解决方案：
= (1/3)-4
= 34 (a-n = 1/an)
= 81
(iv)(-1/2) -1
解决方案：
= (-1/2)-1
= -21 (a-n = 1/an)
= -2
问题3.以p / q的形式将以下各项表示为有理数：
(i)6 -1
解决方案：
= 6-1
= 1/61 = 1/6 (a-n = 1/an)
(ii)(-7) -1
解决方案：
= (-7)-1
= 1/-71 (a-n = 1/an)
= -1/7
(iii)(1/4) -1
解决方案：
= (1/4)-1
= 41 (1/a-n = an)
= 4
(iv)(-4) -1 ×(-3/2) -1
解决方案：
= (-4)-1 × (-3/2)-1
= 1/-41 × (2/-3)1 (a-n = 1/an, 1/a-n = an)
2 is the common factor
= 1/-2 × -1/3
= 1/6
(v)(3/5) -1 ×(5/2) -1
解决方案：
= (3/5)-1 × (5/2)-1(a-n = 1/an)
= (5/3)1 × (2/5)1
= 5/3 × 2/5
= 2/3
问题4.简化：
(i)(4 -1 ×3 -1 ) 2
解决方案：
= (4-1 × 3-1)2
= (1/4 × 1/3)2 (a-n = 1/an)
= (1/12)2
= 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
= 216/125
(iii)(2 -1 + 3 -1 ) -1
解决方案：
= (2-1 + 3-1)-1
= (1/2 + 1/3)-1 (a-n = 1/an)
LCM of 2 and 3 is 6
= ((3+2)/6)-1
= (5/6)-1 (1/a-n = an)
= 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 (1/a-n = an)
= 12 × 1/5
= 12/5
(v)(4 -1 – 5 -1 )÷3 -1
解决方案：
= (4-1 – 5-1) ÷ 3-1
= (1/4 – 1/5) ÷ 1/3 (a-n = 1/an)
LCM of 4 and 5 is 20
= (5 – 4)/20 × 3/1
= 1/20 × 3
= 3/20
问题5.用负指数表示以下每个有理数：
(i)(1/4) 3
解决方案：
= (1/4)3
= (4)-3 (1/an = a-n)
(ii)3 5
解决方案：
= 35
= (1/3)-5 (1/an = a-n)
(iii)(3/5) 4
解决方案：
= (3/5)4
= (5/3)-4 (a/b)-n = (b/a)n
(iv)((3/2) 4 ) -3
解决方案：
= ((3/2)4)-3
= (3/2)-12 ((an)m = anm)
(v)((7/3) 4 ) -3
解决方案：
= ((7/3)4)-3
= (7/3)-12 ((an)m = anm)
问题6.用正指数表示以下每个有理数：
(i)(3/4) -2
解决方案：
= (3/4)-2
= (4/3)2 ((a/b)-n = (b/a)n)
(ii)(5/4) -3
解决方案：
= (5/4)-3
= (4/5)3 ((a/b)-n = (b/a)n)
(iii)4 3 ×4 -9
解决方案：
= 43 × 4-9
= (4)3 – 9 (an × am = an + m)
= 4-6
= (1/4)6 (1/an = a-n)
(iv) ((4/3) -3 ) -4
解决方案：
= ((4/3)-3)-4
= (4/3)12 ((an)m = anm)
(v) ((3/2) 4 ) -2
解决方案：
= ((3/2)4)-2
= (3/2)-8 ((an)m = anm)
= (2/3)8 (1/an = a-n)
问题7：简化：
(i)((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/an = a-n)
= (27-8) ÷ 64
= 19 ÷ 64
= 19/64
(ii)(3 2 – 2 2 )×(2/3) -3
解决方案：
= (32 – 22) × (2/3)-3
= (9 – 4) × (3/2)3 (1/an = a-n)
= 5 × (27/8)
= 135/8
(iii) ((1/2) -1 ×(-4) -1 ) -1
解决方案：
= ((1/2)-1 × (-4)-1)-1
= (21 × (1/-4))-1 (1/an = a-n)
2 is the common factor
= (1/-2)-1 (1/an = a-n)
= -21
= -2
(iv)((((-1/4) 2 ) -2 ) -1
解决方案：
= (((-1/4)2)-2)-1
= ((1/16)-2)-1 (1/an = a-n)
= ((16)2)-1 (1/an = a-n)
= (256)-1 (1/an = a-n)
= 1/256
(v)((2/3) 2 ) 3 ×(1/3) -4 ×3 -1 ×6 -1
解决方案：
= ((2/3)2)3 × (1/3)-4 × 3-1 × 6-1
= (4/9)3 × 34 × 1/3 × 1/6 (1/an = a-n)
= (64/729) × 81 × 1/3 × 1/6
3 is the common factor
= (64/729) × 27 × 1/6
= 32/729 × 27 × 1/3
3 is the common factor
= 32/729 × 9
9 is the common factor
= 32/81
问题8.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 (1/an = a-n)
x = (-1/7) / (1/5)
= (-1/7) × (5/1)
= -5/7
It should be multiplied with -5/7
问题9.(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) (we know that 1/an = a-n)
x = (-7/4) / (2/1)
= (-7/4) × (1/2)
= -7/8
It should be multiplied with -7/8
问题10.(-15) -1应除以多少，以使商等于(-5) -1 ?
解决方案：
Let the number be x
So, (-15)-1 ÷ x = (-5)-1 (we know that 1/a ÷ 1/b = 1/a × b/1)
1/-15 × 1/x = 1/-5 (we know that 1/an = a-n)
1/x = (1× – 15)/-5
1/x = 3
x = 1/3
It should be divided by 1/3