This question appeared in 4GMAT's diagnostic test. This one is a problem solving question from number properties. Concept tested is your understanding of the rules of indices. For integer n > 1, which of the following expressions will have the least value? A. (1/5)^n B. (2)^(-n) C. (10)^(-2n) D. 4^(n/2) E. (0.05)^(-n). Detailed explanation including recap of rules of indices is presented.

1. GMAT QUANTITATIVE REASONING
NUMBER PROPERTIES:
INDICES
PROBLEM SOLVING
Diagnostic Test

2. Question
For integer n > 1, which of the following expressions will have
the least value?
A.
1
5
𝑛
B. 2-n
C. 10-2n
D. 4
𝑛
2
E. (0.05)-n

3. Part 1
Theory Recap : Important Rules relating
to indices

4. Indices Rules
Rule Example

5. Indices Rules
𝑎 𝑥
× 𝑎 𝑦
= 𝑎 𝑥+𝑦
Rule Example

6. Indices Rules
𝑎 𝑥
× 𝑎 𝑦
= 𝑎 𝑥+𝑦
102
× 103
= 105
Rule Example

7. Indices Rules
𝑎 𝑥
× 𝑎 𝑦
= 𝑎 𝑥+𝑦
102
× 103
= 105
Rule Example
𝑎 𝑥 𝑦 = 𝑎 𝑥𝑦

8. Indices Rules
𝑎 𝑥
× 𝑎 𝑦
= 𝑎 𝑥+𝑦
102
× 103
= 105
Rule Example
𝑎 𝑥 𝑦 = 𝑎 𝑥𝑦 102 3
= 106

9. Indices Rules
𝑎 𝑥
× 𝑎 𝑦
= 𝑎 𝑥+𝑦
102
× 103
= 105
Rule Example
𝑎 𝑥 𝑦 = 𝑎 𝑥𝑦 102 3
= 106
𝑎 𝑥
𝑎 𝑦 = 𝑎 𝑥−𝑦

10. Indices Rules
𝑎 𝑥
× 𝑎 𝑦
= 𝑎 𝑥+𝑦
102
× 103
= 105
Rule Example
𝑎 𝑥 𝑦 = 𝑎 𝑥𝑦 102 3
= 106
𝑎 𝑥
𝑎 𝑦 = 𝑎 𝑥−𝑦 105
103
= 102

11. Indices Rules
𝑎 𝑥
× 𝑎 𝑦
= 𝑎 𝑥+𝑦
102
× 103
= 105
Rule Example
𝑎 𝑥 𝑦 = 𝑎 𝑥𝑦 102 3
= 106
𝑎 𝑥
𝑎 𝑦 = 𝑎 𝑥−𝑦 105
103
= 102
𝑎 𝑥
1
𝑦 = 𝑎
𝑥
𝑦 =
𝑦
𝑎 𝑥

12. Indices Rules
𝑎 𝑥
× 𝑎 𝑦
= 𝑎 𝑥+𝑦
102
× 103
= 105
Rule Example
𝑎 𝑥 𝑦 = 𝑎 𝑥𝑦 102 3
= 106
𝑎 𝑥
𝑎 𝑦 = 𝑎 𝑥−𝑦 105
103
= 102
𝑎 𝑥
1
𝑦 = 𝑎
𝑥
𝑦 =
𝑦
𝑎 𝑥 106
1
3 = 10
6
3 = 102

13. Indices Rules
𝑎 𝑥
× 𝑎 𝑦
= 𝑎 𝑥+𝑦
102
× 103
= 105
Rule Example
𝑎 𝑥 𝑦 = 𝑎 𝑥𝑦 102 3
= 106
𝑎 𝑥
𝑎 𝑦 = 𝑎 𝑥−𝑦 105
103
= 102
𝑎 𝑥
1
𝑦 = 𝑎
𝑥
𝑦 =
𝑦
𝑎 𝑥 106
1
3 = 10
6
3 = 102
106
1
3 =
3
106 = 1024th rule can also be
expressed as

14. Indices Rules
𝑎 𝑥
× 𝑎 𝑦
= 𝑎 𝑥+𝑦
102
× 103
= 105
Rule Example
𝑎 𝑥 𝑦 = 𝑎 𝑥𝑦 102 3
= 106
𝑎 𝑥
𝑎 𝑦 = 𝑎 𝑥−𝑦 105
103
= 102
𝑎 𝑥
1
𝑦 = 𝑎
𝑥
𝑦 =
𝑦
𝑎 𝑥
Rule Example
106
1
3 = 10
6
3 = 102
106
1
3 =
3
106 = 1024th rule can also be
expressed as

15. Indices Rules
𝑎 𝑥
× 𝑎 𝑦
= 𝑎 𝑥+𝑦
102
× 103
= 105
Rule Example
𝑎 𝑥 𝑦 = 𝑎 𝑥𝑦 102 3
= 106
𝑎 𝑥
𝑎 𝑦 = 𝑎 𝑥−𝑦 105
103
= 102
𝑎 𝑥
1
𝑦 = 𝑎
𝑥
𝑦 =
𝑦
𝑎 𝑥
Rule Example
106
1
3 = 10
6
3 = 102
𝑎−𝑥
=
1
𝑎 𝑥
106
1
3 =
3
106 = 1024th rule can also be
expressed as

16. Indices Rules
𝑎 𝑥
× 𝑎 𝑦
= 𝑎 𝑥+𝑦
102
× 103
= 105
Rule Example
𝑎 𝑥 𝑦 = 𝑎 𝑥𝑦 102 3
= 106
𝑎 𝑥
𝑎 𝑦 = 𝑎 𝑥−𝑦 105
103
= 102
𝑎 𝑥
1
𝑦 = 𝑎
𝑥
𝑦 =
𝑦
𝑎 𝑥
Rule Example
106
1
3 = 10
6
3 = 102
𝑎−𝑥
=
1
𝑎 𝑥
2−3
=
1
23
106
1
3 =
3
106 = 1024th rule can also be
expressed as

17. Indices Rules
𝑎 𝑥
× 𝑎 𝑦
= 𝑎 𝑥+𝑦
102
× 103
= 105
Rule Example
𝑎 𝑥 𝑦 = 𝑎 𝑥𝑦 102 3
= 106
𝑎 𝑥
𝑎 𝑦 = 𝑎 𝑥−𝑦 105
103
= 102
𝑎 𝑥
1
𝑦 = 𝑎
𝑥
𝑦 =
𝑦
𝑎 𝑥
Rule Example
106
1
3 = 10
6
3 = 102
𝑎−𝑥
=
1
𝑎 𝑥
2−3
=
1
23
𝑎 𝑥
× 𝑏 𝑥
= 𝑎𝑏 𝑥
106
1
3 =
3
106 = 1024th rule can also be
expressed as

18. Indices Rules
𝑎 𝑥
× 𝑎 𝑦
= 𝑎 𝑥+𝑦
102
× 103
= 105
Rule Example
𝑎 𝑥 𝑦 = 𝑎 𝑥𝑦 102 3
= 106
𝑎 𝑥
𝑎 𝑦 = 𝑎 𝑥−𝑦 105
103
= 102
𝑎 𝑥
1
𝑦 = 𝑎
𝑥
𝑦 =
𝑦
𝑎 𝑥
Rule Example
106
1
3 = 10
6
3 = 102
𝑎−𝑥
=
1
𝑎 𝑥
2−3
=
1
23
𝑎 𝑥
× 𝑏 𝑥
= 𝑎𝑏 𝑥
23
× 33
= 63
106
1
3 =
3
106 = 1024th rule can also be
expressed as

19. Part 2
Apply the Rules to solve the question

20. Which will have the least value?
A.
1
5
𝑛
B. 2-n C. 10-2n D. 4
𝑛
2 E. (0.05)-n

21. Which will have the least value?
A.
1
5
𝑛
B. 2-n C. 10-2n D. 4
𝑛
2 E. (0.05)-n
All of the answer choices have an ‘n’ term in their index.

22. Which will have the least value?
Step 1: To make comparison meaningful rewrite all expressions to have the same power
A.
1
5
𝑛
B. 2-n C. 10-2n D. 4
𝑛
2 E. (0.05)-n
All of the answer choices have an ‘n’ term in their index.

23. Which will have the least value?
Step 1: To make comparison meaningful rewrite all expressions to have the same power
A.
1
5
𝑛
B. 2-n C. 10-2n D. 4
𝑛
2 E. (0.05)-n
All of the answer choices have an ‘n’ term in their index.
1
5
𝑛
A.

24. Which will have the least value?
Step 1: To make comparison meaningful rewrite all expressions to have the same power
A.
1
5
𝑛
B. 2-n C. 10-2n D. 4
𝑛
2 E. (0.05)-n
All of the answer choices have an ‘n’ term in their index.
1
5
𝑛
The power is ‘n’. So, nothing needs to be doneA.

25. Which will have the least value?
Step 1: To make comparison meaningful rewrite all expressions to have the same power
A.
1
5
𝑛
B. 2-n C. 10-2n D. 4
𝑛
2 E. (0.05)-n
All of the answer choices have an ‘n’ term in their index.
1
5
𝑛
The power is ‘n’. So, nothing needs to be done
B. 2-n
A.

26. Which will have the least value?
Step 1: To make comparison meaningful rewrite all expressions to have the same power
A.
1
5
𝑛
B. 2-n C. 10-2n D. 4
𝑛
2 E. (0.05)-n
All of the answer choices have an ‘n’ term in their index.
1
5
𝑛
The power is ‘n’. So, nothing needs to be done
B. 2-n Rule: 𝑎−𝑥 =
1
𝑎 𝑥 .
A.

27. Which will have the least value?
Step 1: To make comparison meaningful rewrite all expressions to have the same power
A.
1
5
𝑛
B. 2-n C. 10-2n D. 4
𝑛
2 E. (0.05)-n
All of the answer choices have an ‘n’ term in their index.
1
5
𝑛
The power is ‘n’. So, nothing needs to be done
B. 2-n Rule: 𝑎−𝑥 =
1
𝑎 𝑥 . So, 2−𝑛
=
1
2 𝑛 =
1
2
𝑛
A.

28. Which will have the least value?
Step 1: To make comparison meaningful rewrite all expressions to have the same power
A.
1
5
𝑛
B. 2-n C. 10-2n D. 4
𝑛
2 E. (0.05)-n
All of the answer choices have an ‘n’ term in their index.
1
5
𝑛
The power is ‘n’. So, nothing needs to be done
B. 2-n Rule: 𝑎−𝑥 =
1
𝑎 𝑥 . So, 2−𝑛
=
1
2 𝑛 =
1
2
𝑛
C. 10-2n
A.

29. Which will have the least value?
Step 1: To make comparison meaningful rewrite all expressions to have the same power
A.
1
5
𝑛
B. 2-n C. 10-2n D. 4
𝑛
2 E. (0.05)-n
All of the answer choices have an ‘n’ term in their index.
1
5
𝑛
The power is ‘n’. So, nothing needs to be done
B. 2-n Rule: 𝑎−𝑥 =
1
𝑎 𝑥 . So, 2−𝑛
=
1
2 𝑛 =
1
2
𝑛
C. 10-2n Rule: 𝑎−𝑥
=
1
𝑎 𝑥 .
A.

30. Which will have the least value?
Step 1: To make comparison meaningful rewrite all expressions to have the same power
A.
1
5
𝑛
B. 2-n C. 10-2n D. 4
𝑛
2 E. (0.05)-n
All of the answer choices have an ‘n’ term in their index.
1
5
𝑛
The power is ‘n’. So, nothing needs to be done
B. 2-n Rule: 𝑎−𝑥 =
1
𝑎 𝑥 . So, 2−𝑛
=
1
2 𝑛 =
1
2
𝑛
C. 10-2n Rule: 𝑎−𝑥
=
1
𝑎 𝑥 . So, 10−2𝑛 =
1
102𝑛 =
1
102
𝑛
.
A.

31. Which will have the least value?
Step 1: To make comparison meaningful rewrite all expressions to have the same power
A.
1
5
𝑛
B. 2-n C. 10-2n D. 4
𝑛
2 E. (0.05)-n
All of the answer choices have an ‘n’ term in their index.
1
5
𝑛
The power is ‘n’. So, nothing needs to be done
B. 2-n Rule: 𝑎−𝑥 =
1
𝑎 𝑥 . So, 2−𝑛
=
1
2 𝑛 =
1
2
𝑛
C. 10-2n Rule: 𝑎−𝑥
=
1
𝑎 𝑥 . So, 10−2𝑛 =
1
102𝑛 =
1
102
𝑛
. Which is
1
100
𝑛
A.

32. Which will have the least value?
Step 1: To make comparison meaningful rewrite all expressions to have the same power
A.
1
5
𝑛
B. 2-n C. 10-2n D. 4
𝑛
2 E. (0.05)-n
All of the answer choices have an ‘n’ term in their index.
1
5
𝑛
The power is ‘n’. So, nothing needs to be done
B. 2-n Rule: 𝑎−𝑥 =
1
𝑎 𝑥 . So, 2−𝑛
=
1
2 𝑛 =
1
2
𝑛
C. 10-2n Rule: 𝑎−𝑥
=
1
𝑎 𝑥 . So, 10−2𝑛 =
1
102𝑛 =
1
102
𝑛
. Which is
1
100
𝑛
A.
D. 4
𝑛
2

33. Which will have the least value?
Step 1: To make comparison meaningful rewrite all expressions to have the same power
A.
1
5
𝑛
B. 2-n C. 10-2n D. 4
𝑛
2 E. (0.05)-n
All of the answer choices have an ‘n’ term in their index.
1
5
𝑛
The power is ‘n’. So, nothing needs to be done
B. 2-n Rule: 𝑎−𝑥 =
1
𝑎 𝑥 . So, 2−𝑛
=
1
2 𝑛 =
1
2
𝑛
C. 10-2n Rule: 𝑎−𝑥
=
1
𝑎 𝑥 . So, 10−2𝑛 =
1
102𝑛 =
1
102
𝑛
. Which is
1
100
𝑛
A.
D. 4
𝑛
2 Rule: 𝑎 𝑥
1
𝑦 = 𝑎
𝑥
𝑦 =
𝑦
𝑎 𝑥.

34. Which will have the least value?
Step 1: To make comparison meaningful rewrite all expressions to have the same power
A.
1
5
𝑛
B. 2-n C. 10-2n D. 4
𝑛
2 E. (0.05)-n
All of the answer choices have an ‘n’ term in their index.
1
5
𝑛
The power is ‘n’. So, nothing needs to be done
B. 2-n Rule: 𝑎−𝑥 =
1
𝑎 𝑥 . So, 2−𝑛
=
1
2 𝑛 =
1
2
𝑛
C. 10-2n Rule: 𝑎−𝑥
=
1
𝑎 𝑥 . So, 10−2𝑛 =
1
102𝑛 =
1
102
𝑛
. Which is
1
100
𝑛
A.
D. 4
𝑛
2 Rule: 𝑎 𝑥
1
𝑦 = 𝑎
𝑥
𝑦 =
𝑦
𝑎 𝑥. So, 4
𝑛
2 =
2
4
𝑛
= 2 𝑛

35. Which will have the least value?
Step 1: To make comparison meaningful rewrite all expressions to have the same power
A.
1
5
𝑛
B. 2-n C. 10-2n D. 4
𝑛
2 E. (0.05)-n
All of the answer choices have an ‘n’ term in their index.
1
5
𝑛
The power is ‘n’. So, nothing needs to be done
B. 2-n Rule: 𝑎−𝑥 =
1
𝑎 𝑥 . So, 2−𝑛
=
1
2 𝑛 =
1
2
𝑛
C. 10-2n Rule: 𝑎−𝑥
=
1
𝑎 𝑥 . So, 10−2𝑛 =
1
102𝑛 =
1
102
𝑛
. Which is
1
100
𝑛
A.
D. 4
𝑛
2 Rule: 𝑎 𝑥
1
𝑦 = 𝑎
𝑥
𝑦 =
𝑦
𝑎 𝑥. So, 4
𝑛
2 =
2
4
𝑛
= 2 𝑛
E. (0.05)-n

36. Which will have the least value?
Step 1: To make comparison meaningful rewrite all expressions to have the same power
A.
1
5
𝑛
B. 2-n C. 10-2n D. 4
𝑛
2 E. (0.05)-n
All of the answer choices have an ‘n’ term in their index.
1
5
𝑛
The power is ‘n’. So, nothing needs to be done
B. 2-n Rule: 𝑎−𝑥 =
1
𝑎 𝑥 . So, 2−𝑛
=
1
2 𝑛 =
1
2
𝑛
C. 10-2n Rule: 𝑎−𝑥
=
1
𝑎 𝑥 . So, 10−2𝑛 =
1
102𝑛 =
1
102
𝑛
. Which is
1
100
𝑛
A.
D. 4
𝑛
2 Rule: 𝑎 𝑥
1
𝑦 = 𝑎
𝑥
𝑦 =
𝑦
𝑎 𝑥. So, 4
𝑛
2 =
2
4
𝑛
= 2 𝑛
E. (0.05)-n =
5
100
−𝑛

37. Which will have the least value?
Step 1: To make comparison meaningful rewrite all expressions to have the same power
A.
1
5
𝑛
B. 2-n C. 10-2n D. 4
𝑛
2 E. (0.05)-n
All of the answer choices have an ‘n’ term in their index.
1
5
𝑛
The power is ‘n’. So, nothing needs to be done
B. 2-n Rule: 𝑎−𝑥 =
1
𝑎 𝑥 . So, 2−𝑛
=
1
2 𝑛 =
1
2
𝑛
C. 10-2n Rule: 𝑎−𝑥
=
1
𝑎 𝑥 . So, 10−2𝑛 =
1
102𝑛 =
1
102
𝑛
. Which is
1
100
𝑛
A.
D. 4
𝑛
2 Rule: 𝑎 𝑥
1
𝑦 = 𝑎
𝑥
𝑦 =
𝑦
𝑎 𝑥. So, 4
𝑛
2 =
2
4
𝑛
= 2 𝑛
E. (0.05)-n =
5
100
−𝑛
=
1
20
−𝑛
.

38. Which will have the least value?
Step 1: To make comparison meaningful rewrite all expressions to have the same power
A.
1
5
𝑛
B. 2-n C. 10-2n D. 4
𝑛
2 E. (0.05)-n
All of the answer choices have an ‘n’ term in their index.
1
5
𝑛
The power is ‘n’. So, nothing needs to be done
B. 2-n Rule: 𝑎−𝑥 =
1
𝑎 𝑥 . So, 2−𝑛
=
1
2 𝑛 =
1
2
𝑛
C. 10-2n Rule: 𝑎−𝑥
=
1
𝑎 𝑥 . So, 10−2𝑛 =
1
102𝑛 =
1
102
𝑛
. Which is
1
100
𝑛
A.
D. 4
𝑛
2 Rule: 𝑎 𝑥
1
𝑦 = 𝑎
𝑥
𝑦 =
𝑦
𝑎 𝑥. So, 4
𝑛
2 =
2
4
𝑛
= 2 𝑛
E. (0.05)-n =
5
100
−𝑛
=
1
20
−𝑛
. Rule: 𝑎−𝑥 =
1
𝑎 𝑥 .

39. Which will have the least value?
Step 1: To make comparison meaningful rewrite all expressions to have the same power
A.
1
5
𝑛
B. 2-n C. 10-2n D. 4
𝑛
2 E. (0.05)-n
All of the answer choices have an ‘n’ term in their index.
1
5
𝑛
The power is ‘n’. So, nothing needs to be done
B. 2-n Rule: 𝑎−𝑥 =
1
𝑎 𝑥 . So, 2−𝑛
=
1
2 𝑛 =
1
2
𝑛
C. 10-2n Rule: 𝑎−𝑥
=
1
𝑎 𝑥 . So, 10−2𝑛 =
1
102𝑛 =
1
102
𝑛
. Which is
1
100
𝑛
A.
D. 4
𝑛
2 Rule: 𝑎 𝑥
1
𝑦 = 𝑎
𝑥
𝑦 =
𝑦
𝑎 𝑥. So, 4
𝑛
2 =
2
4
𝑛
= 2 𝑛
E. (0.05)-n =
5
100
−𝑛
=
1
20
−𝑛
. Rule: 𝑎−𝑥 =
1
𝑎 𝑥 . So,
1
20
−𝑛
= 20n

40. Which will have the least value?
A.
1
5
𝑛
B. 2-n C. 10-2n D. 4
𝑛
2 E. (0.05)-n

41. Step 2: Now that we have expressed all choices to power ‘n’, just compare the bases
Which will have the least value?
A.
1
5
𝑛
B. 2-n C. 10-2n D. 4
𝑛
2 E. (0.05)-n

42. Step 2: Now that we have expressed all choices to power ‘n’, just compare the bases
Which will have the least value?
A.
1
5
𝑛
B. 2-n C. 10-2n D. 4
𝑛
2 E. (0.05)-n
Listing down only the bases for all 5 choices

43. Step 2: Now that we have expressed all choices to power ‘n’, just compare the bases
Which will have the least value?
A.
1
5
𝑛
B. 2-n C. 10-2n D. 4
𝑛
2 E. (0.05)-n
Listing down only the bases for all 5 choices
A.
1
5
B.
1
2
C.
1
100
D. 2 E. 20

44. Step 2: Now that we have expressed all choices to power ‘n’, just compare the bases
Which will have the least value?
A.
1
5
𝑛
B. 2-n C. 10-2n D. 4
𝑛
2 E. (0.05)-n
Listing down only the bases for all 5 choices
A.
1
5
B.
1
2
C.
1
100
D. 2 E. 20
Of the 5 choices, the smallest number is
1
100

45. Step 2: Now that we have expressed all choices to power ‘n’, just compare the bases
Which will have the least value?
A.
1
5
𝑛
B. 2-n C. 10-2n D. 4
𝑛
2 E. (0.05)-n
Listing down only the bases for all 5 choices
A.
1
5
B.
1
2
C.
1
100
D. 2 E. 20
Of the 5 choices, the smallest number is
1
100
10-2n is the least value

46. Step 2: Now that we have expressed all choices to power ‘n’, just compare the bases
Which will have the least value?
A.
1
5
𝑛
B. 2-n C. 10-2n D. 4
𝑛
2 E. (0.05)-n
Listing down only the bases for all 5 choices
A.
1
5
B.
1
2
C.
1
100
D. 2 E. 20
Of the 5 choices, the smallest number is
1
100
Choices C is the answer.
10-2n is the least value

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