1. The document contains examples of exponent rules and operations with positive and negative numbers raised to powers. 2. Steps are shown to simplify expressions using exponent rules like distributing exponents across multiplication and rewriting repeated multiplication as exponents. 3. Various examples calculate integer and decimal values when numbers are raised to powers.

1. 1. ʿ F ʾ 1
F ˈ 3 F F
ʿ F
ʿ F
F
ʿ F
2. ʿ ˈ ʿ F 1
3. ʿ F F F
3.1 F
3.2
3.3 ʿ F F F
3.4 ʿ F F F F
3.5
3.6
3.7
3.8 F F ʿ

2. F 1
F × F
1. an
F
. a × a × a × × a ( n ) . a + a + a + + a ( n )
. n + n + n + + n ( n ) . n × n × n × n ( n )
2. (-7) ×(-7) × (-7) × (-7) F F F F
. (-7) . (-7)7
. (-7)4
. - 74
3. 32
. 2 . 3
. 23
. 32
4. - 23
F F F
. .
. .
5. ( b2
)3
F
. b2
× b2
× b2
. b2
+ b2
+ b2
. b3
× b3
× b3
. b3
+ b3
+ b3
6. 0.000729 F F F
. (0.9) 3
. (0.09) 3
. (0.009) 3
. (0.003)9
7. 0.0000001 F F ˈ 10 F F
. 103
. 107
. 10 6−
. 10 7−

3. 8. 5n
= 625 , n F
. 2 . 3
. 4 . 5
9. 81 F F
. 81 = 2 × 2 × 2 × 2 . 81 = 3 × 3 × 3 × 3
. 81 = 4 × 4 × 4 × 4 . 81 = 5 × 5 × 5 × 5
10. (0.2)3
F F F
. 0.008 . 0.0008
. 0.08 . 0.8
******************

4. F 1
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.

5. ʿ F 1
F F
1. F
2. ˈ
F F

6. F a n ˈ F
an
= a × a × a × × a
n
an
F
a F
n F
F
F 54
54
F F F
5 F
4 F
F (-2 )4
(-2 ) 4
F F
(-2 ) F
4 F
F -25
-2
5
F F F
2 F
5 F
F 





5
1 6






5
1 6
F F F F






5
1
F
6 F

7. F a n ˈ F
an
= a × a × a × × a
n
an
ˈ a ˈ n ˈ
an
F
F

8. ʿ F
1.1
F F F
F 36
F F
3 F
6 F
1. 73
F F
............ F
............ F
2. (-5 )4
F F
.......... F
.......... F
3.
5
2
1






F F
2
1
F
5 F
4. (0.4) 3
F F
(0.4) F
3 F
5. 3 2
F F .
3 F
2 F

9. ʿ F
1.1
F F F
1. 73
F F
7 F
3 F
2. (-5 )4
F F F
-5 F
4 F
3.
5
2
1






F F F F
2
1
F
5 F
4. (0.4) 3
F F F
(0.4) F
3 F
5. 32
F F
3 F
2 F

10. a n
an
= a × a × a × × a a n
n
F ˈ F F
F
53
5 × 5 × 5
(-2 )6
(-2 ) × (-2 ) × (-2 ) × (-2 ) × (-2 ) × (-2 )
F
F F 7 × 7 × 7 × 7 × 7 F 75

11. ʿ F
1.2
F F
F 35
= 3 × 3 × 3 × 3 × 3
1. 135
=
2. (- 7 )4
= ...............................................
3. (0.6)3
=
4. (1.2) 3
= ..
5. (-
3
2
)5
=
6. C6
=

12. ʿ F
1.2
1. 135
= 13 × 13 × 13 × 13 × 13
2. (- 7 )4
= (-7 ) × (-7 ) × (-7 ) × (-7 )
3. (0.6)3
= (0.6) × (0.6) × (0.6)
4. (1.2) 3
= (1.2) × (1.2) × (1.2)
5. (-
3
2
)5
= (-
3
2
) × (-
3
2
) × (-
3
2
) × (-
3
2
) × (-
3
2
)
6. C6
= C × C × C × C × C × C

13. ˈ
104
= 10 × 10 × 10 × 10 = 10,000
103
= 10 × 10 × 10 = 1,000
102
= 10 × 10 = 100
101
= 10
100
= 1
102
F F
y
10
0 10 x
102
F F
= 102
F
ˈ F × F × F
103
F F F F
y
10
10
z 0 10 x
103
F F
F = 103
F
F × ×

14. ʿ F
1.3
F F F F F F F F
............... 1. 10 × 10× 10 × 10 × 10 × 10 . 107
2. 10 × 10× 10 × 10 . 108
3. 10 × 10× 10 . 101
4. 10 × 10× 10 × 10 × 10 × 10 × 10 . 10 1−n
5. 10 × 10 . 10 6
6. 10 × 10× 10 × 10 × 10 × 10 × 10 × 10 . 109
7. 10 . 103
8. 10 × 10× 10 × 10 × 10 × 10 × 10 × 10 × 10 . 104
9. 1 . 100
...10. 10 × 10× 10 ×...× 10 . 102
n 1 . 105
. 1010
. 10n

15. ʿ F
1.3
1. 10 × 10× 10 × 10 × 10 × 10
2. 10 × 10× 10 × 10
3. 10 × 10× 10
4. 10 × 10× 10 × 10 × 10 × 10 × 10
5. 10 × 10
6. 10 × 10× 10 × 10 × 10 × 10 × 10 × 10
7. 10
8. 10 × 10× 10 × 10 × 10 × 10 × 10 × 10 × 10
9. 1
10. 10 × 10× 10 ×...× 10
n 1

16. ʿ F
1.4
F F F
F 10 4
= 10 × 10 × 10 × 10 = 10,000
1. 103
= = .
2. 105
= = .
3. 107
= = .
4. 109
= = .
5. 102
= = .
6. 106
= =
7. 108
= = .
8. 1010
= =
9. 10 1+n
= = .
10. 10 2−n
= = ..

17. ʿ F
1.4
1. 103
= 10 × 10× 10 = 1,000
2. 105
= 10 × 10× 10 × 10 × 10 = 10,000
3. 107
= 10 × 10× 10 × 10 × 10 × 10 × 10 = 10,000,000
4. 109
= 10 × 10× 10 × 10 × 10 × 10 × 10 × 10 × 10 = 1,000,000,000
5. 102
= 10 × 10 = 100
6. 106
= 10 × 10× 10 × 10 × 10 × 10 = 1,000,000
7. 108
= 10 × 10× 10 × 10 × 10 × 10 × 10 × 10 = 100,000,000
8. 1010
= 10 × 10× 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 = 10,000,000,000
9. 10 1+n
= 10 × 10× 10 ×...× 10
n +1
10. 10 2−n
= 10 × 10× 10 ×...× 10
n -2

18. ʿ F
1.5
F
ˈ
F 1) 10,000,000,000
1 × 1010
2) F ʾ . . 2548
42,000,000
42 × 106
1. F F 1,000,000
2. ˈ ˁ F 2,400,000 F
3. F F F 100,000,000,000
4. F F 39,000
5. F F F 93,000,000 F

19. ʿ F
1.5
1. F F 1,000,000
1× 106
2. ˈ ˁ F 2,400,000 F
24 × 105
3. F F F 100,000,000,000
1×1011
4. F F 39,000
39 × 103
5. F F F 93,000,000 F
93 × 106

20. ˈ
F a ˈ n ˈ F
a n
= a n
a ˈ F a ˈ
F F
F 34
= 3 × 3 × 3 × 3
(-2 )3
= (-2) × (-2 ) × (-2 )
3
3
2





 x
= 





3
2x
× 





3
2x
× 





3
2x
(0.5)2
= (0.5) × (0.5)

21. ʿ F
1.6
F F F
F (-2) × (-2 ) × (-2 ) × (-2 ) × (-2 ) = (- 2 )5
1. 7 × 7 × 7 × 7 × 7 × 7 =
2.
5
4
×
5
4
×
5
4
=
3. 




−
8
3
× 




−
8
3
× 




−
8
3
× 




−
8
3
=
4. (-0.7) × (-0.7) ×(-0.7) =
5. - 




 ×××
9
5
9
5
9
5
9
5
=
6. ab × ab × ab =

22. ʿ F
1.6
1. 7 × 7 × 7 × 7 × 7 × 7 = 76
2.
5
4
×
5
4
×
5
4
=
3
5
4






3. 




−
8
3
× 




−
8
3
× 




−
8
3
× 




−
8
3
=
4
8
3





−
4. (-0.7) × (-0.7) ×(-0.7) = (-0.7)3
5. - 




 ×××
9
5
9
5
9
5
9
5
= -
4
9
5






6. ab × ab × ab = (ab)3

23. F F
F F F
F F F
F F F F
27 = 3 × 3 × 3 = 33
-8 = (-2 ) × (-2 ) × (-2 ) = (-2 )3
0.008 = (0.2) × (0.2) × (0.2) = (0.2)3
16 2 × 2 × 2 × 2 24
16 (-2 )×(-2 )×(-2 )×(-2 ) (-2 )4
16 4 ×4 42
16 (-4 ) × (-4 ) (-4 )2

24. ʿ F
1.7
F F F F
F 64 = 2 × 2 × 2 × 2 × 2 × 2
= 26
1. 243 = ..
=
2. -64 = ..
= ..
3. 0.0625 = ..
=
4. 1,000 = ..
=
5.
32
9
= ..
=
=

25. ʿ F
1.7
1. 243 = 3 × 3 × 3 × 3 × 3
= 35
3. -64 = (-4 ) × (-4 ) × (-4 ) - (4 × 4 × 4)
= (-4 )3
- 43
4. 0.0625 = (0.5) × (0.5) × (0.5) ×(0.5)
= (0.5)4
5. 1,000 = 10 × 10 × 10
= 103
6.
32
9
=
22222
33
××××
×
= 5
2
2
3

26. ʿ F
1.8
F F F F F
F 23
= 2 × 2 × 2 = 8
1. 33
= . = .
2. 44
= . =
3. 53
= .=
4. 74
= ................................ = .
5. 103
= ................................... = ..

27. ʿ F
1.8
F F F F F
1. 33
= 3 × 3 × 3 = 27
2. 44
= 4 × 4 × 4 × × = 256
3. 53
= 5 × 5 × 5 = 125
4. 74
= 7 × 7 × 7 × 7 = 2,401
5. 103
= 10 × 10 × 10 = 1,000

28. ʿ F
1.9
F F F F
52
F
25
3 2
5
42
72
82
112
312
212
52

29. ʿ F
1.9
F F F F
42
72
1
82
6 4 112
312
9 6 1
212
4 4 1
52
2 5
1

30. F F ʿ F 1
F × F F
1. (b2
)3
F
. b2
× b2
× b2
. b2
+ b2
+ b2
. b3
× b3
× b3
. b3
+ b3
+ b3
2. 32
. 2 . 3
. 23
. 32
3. 0.0000001 F F F F
. 100
. 107
. 10 6−
. 10 7−
4. 81 F F
. 2 × 2 × 2 × 2 . 3 × 3 × 3 × 3
. 4 × 4 × 4 × 4 . 9 × 9
5. mn
F
. m × m × m × ×m (n ) . m + m + m + + m ( n )
. n × n × n × ×n ( n ) . n + n + n + + n ( n )
6. 5n
= 625 , n F
. 2 . 3
. 4 . 5
7. (-7 ) × (-7 ) × (-7 ) ×(-7 ) F F
. (-7 ) . (-7 )7
. (-7 ) 4
. - 74

31. 8. (0.2)3
F F F
. 0.008 . 0.0008
. 0.08 . 0.8
9. - 23
F F F
. .
. .
10. 0.000729 F F F
. (0.9)3
. (0.09)3
. (0.009)3
. (0.003)9

32. F F ʿ F 1
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.