GMAT QUANTITATIVE REASONING

NUMBER PROPERTIES:

INDICES

PROBLEM SOLVING

Diagnostic Test

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. 4𝑛

2

E. (0.05)-n

Part 1

Theory Recap : Important Rules relating

to indices

Indices Rules

Rule Example

Indices Rules

𝑎𝑥 × 𝑎𝑦 = 𝑎𝑥+𝑦

Rule Example

Indices Rules

𝑎𝑥 × 𝑎𝑦 = 𝑎𝑥+𝑦 102 × 103 = 105

Rule Example

Indices Rules

𝑎𝑥 × 𝑎𝑦 = 𝑎𝑥+𝑦 102 × 103 = 105

Rule Example

𝑎𝑥 𝑦 = 𝑎𝑥𝑦

Indices Rules

𝑎𝑥 × 𝑎𝑦 = 𝑎𝑥+𝑦 102 × 103 = 105

Rule Example

𝑎𝑥 𝑦 = 𝑎𝑥𝑦 102 3 = 106

Indices Rules

𝑎𝑥 × 𝑎𝑦 = 𝑎𝑥+𝑦 102 × 103 = 105

Rule Example

𝑎𝑥 𝑦 = 𝑎𝑥𝑦 102 3 = 106

𝑎𝑥

𝑎𝑦= 𝑎𝑥−𝑦

Indices Rules

𝑎𝑥 × 𝑎𝑦 = 𝑎𝑥+𝑦 102 × 103 = 105

Rule Example

𝑎𝑥 𝑦 = 𝑎𝑥𝑦 102 3 = 106

𝑎𝑥

𝑎𝑦= 𝑎𝑥−𝑦

105

103= 102

Indices Rules

𝑎𝑥 × 𝑎𝑦 = 𝑎𝑥+𝑦 102 × 103 = 105

Rule Example

𝑎𝑥 𝑦 = 𝑎𝑥𝑦 102 3 = 106

𝑎𝑥

𝑎𝑦= 𝑎𝑥−𝑦

105

103= 102

𝑎𝑥1𝑦 = 𝑎

𝑥𝑦 =

𝑦𝑎𝑥

Indices Rules

𝑎𝑥 × 𝑎𝑦 = 𝑎𝑥+𝑦 102 × 103 = 105

Rule Example

𝑎𝑥 𝑦 = 𝑎𝑥𝑦 102 3 = 106

𝑎𝑥

𝑎𝑦= 𝑎𝑥−𝑦

105

103= 102

𝑎𝑥1𝑦 = 𝑎

𝑥𝑦 =

𝑦𝑎𝑥 106

13 = 10

63 = 102

Indices Rules

𝑎𝑥 × 𝑎𝑦 = 𝑎𝑥+𝑦 102 × 103 = 105

Rule Example

𝑎𝑥 𝑦 = 𝑎𝑥𝑦 102 3 = 106

𝑎𝑥

𝑎𝑦= 𝑎𝑥−𝑦

105

103= 102

𝑎𝑥1𝑦 = 𝑎

𝑥𝑦 =

𝑦𝑎𝑥 106

13 = 10

63 = 102

10613 =

3106 = 102

4th rule can also be

expressed as

Indices Rules

𝑎𝑥 × 𝑎𝑦 = 𝑎𝑥+𝑦 102 × 103 = 105

Rule Example

𝑎𝑥 𝑦 = 𝑎𝑥𝑦 102 3 = 106

𝑎𝑥

𝑎𝑦= 𝑎𝑥−𝑦

105

103= 102

𝑎𝑥1𝑦 = 𝑎

𝑥𝑦 =

𝑦𝑎𝑥

Rule Example

10613 = 10

63 = 102

10613 =

3106 = 102

4th rule can also be

expressed as

Indices Rules

𝑎𝑥 × 𝑎𝑦 = 𝑎𝑥+𝑦 102 × 103 = 105

Rule Example

𝑎𝑥 𝑦 = 𝑎𝑥𝑦 102 3 = 106

𝑎𝑥

𝑎𝑦= 𝑎𝑥−𝑦

105

103= 102

𝑎𝑥1𝑦 = 𝑎

𝑥𝑦 =

𝑦𝑎𝑥

Rule Example

10613 = 10

63 = 102

𝑎−𝑥 =1

𝑎𝑥

10613 =

3106 = 102

4th rule can also be

expressed as

Indices Rules

𝑎𝑥 × 𝑎𝑦 = 𝑎𝑥+𝑦 102 × 103 = 105

Rule Example

𝑎𝑥 𝑦 = 𝑎𝑥𝑦 102 3 = 106

𝑎𝑥

𝑎𝑦= 𝑎𝑥−𝑦

105

103= 102

𝑎𝑥1𝑦 = 𝑎

𝑥𝑦 =

𝑦𝑎𝑥

Rule Example

10613 = 10

63 = 102

𝑎−𝑥 =1

𝑎𝑥2−3 =

1

23

10613 =

3106 = 102

4th rule can also be

expressed as

Indices Rules

𝑎𝑥 × 𝑎𝑦 = 𝑎𝑥+𝑦 102 × 103 = 105

Rule Example

𝑎𝑥 𝑦 = 𝑎𝑥𝑦 102 3 = 106

𝑎𝑥

𝑎𝑦= 𝑎𝑥−𝑦

105

103= 102

𝑎𝑥1𝑦 = 𝑎

𝑥𝑦 =

𝑦𝑎𝑥

Rule Example

10613 = 10

63 = 102

𝑎−𝑥 =1

𝑎𝑥2−3 =

1

23

𝑎𝑥 × 𝑏𝑥 = 𝑎𝑏 𝑥

10613 =

3106 = 102

4th rule can also be

expressed as

Indices Rules

𝑎𝑥 × 𝑎𝑦 = 𝑎𝑥+𝑦 102 × 103 = 105

Rule Example

𝑎𝑥 𝑦 = 𝑎𝑥𝑦 102 3 = 106

𝑎𝑥

𝑎𝑦= 𝑎𝑥−𝑦

105

103= 102

𝑎𝑥1𝑦 = 𝑎

𝑥𝑦 =

𝑦𝑎𝑥

Rule Example

10613 = 10

63 = 102

𝑎−𝑥 =1

𝑎𝑥2−3 =

1

23

𝑎𝑥 × 𝑏𝑥 = 𝑎𝑏 𝑥 23 × 33 = 63

10613 =

3106 = 102

4th rule can also be

expressed as

Part 2

Apply the Rules to solve the question

Which will have the least value?A.

1

5

𝑛B. 2-n C. 10-2n D. 4

𝑛

2 E. (0.05)-n

Which will have the least value?A.

1

5

𝑛B. 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.

Which will have the least value?

Step 1: To make comparison meaningful rewrite all expressions to have the same power

A.1

5

𝑛B. 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.

Which will have the least value?

Step 1: To make comparison meaningful rewrite all expressions to have the same power

A.1

5

𝑛B. 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

𝑛A.

Which will have the least value?

Step 1: To make comparison meaningful rewrite all expressions to have the same power

A.1

5

𝑛B. 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

𝑛The power is ‘n’. So, nothing needs to be doneA.

Which will have the least value?

Step 1: To make comparison meaningful rewrite all expressions to have the same power

A.1

5

𝑛B. 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

𝑛The power is ‘n’. So, nothing needs to be done

B. 2-n

A.

Which will have the least value?

Step 1: To make comparison meaningful rewrite all expressions to have the same power

A.1

5

𝑛B. 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

𝑛The power is ‘n’. So, nothing needs to be done

B. 2-n Rule: 𝑎−𝑥 =1

𝑎𝑥.

A.

Which will have the least value?

Step 1: To make comparison meaningful rewrite all expressions to have the same power

A.1

5

𝑛B. 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

𝑛The power is ‘n’. So, nothing needs to be done

B. 2-n Rule: 𝑎−𝑥 =1

𝑎𝑥. So, 2−𝑛 =

1

2𝑛=

1

2

𝑛

A.

Which will have the least value?

Step 1: To make comparison meaningful rewrite all expressions to have the same power

A.1

5

𝑛B. 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

𝑛The power is ‘n’. So, nothing needs to be done

B. 2-n Rule: 𝑎−𝑥 =1

𝑎𝑥. So, 2−𝑛 =

1

2𝑛=

1

2

𝑛

C. 10-2n

A.

Which will have the least value?

Step 1: To make comparison meaningful rewrite all expressions to have the same power

A.1

5

𝑛B. 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

𝑛The power is ‘n’. So, nothing needs to be done

B. 2-n Rule: 𝑎−𝑥 =1

𝑎𝑥. So, 2−𝑛 =

1

2𝑛=

1

2

𝑛

C. 10-2n Rule: 𝑎−𝑥 =1

𝑎𝑥.

A.

Which will have the least value?

Step 1: To make comparison meaningful rewrite all expressions to have the same power

A.1

5

𝑛B. 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

𝑛The power is ‘n’. So, nothing needs to be done

B. 2-n Rule: 𝑎−𝑥 =1

𝑎𝑥. So, 2−𝑛 =

1

2𝑛=

1

2

𝑛

C. 10-2n Rule: 𝑎−𝑥 =1

𝑎𝑥. So, 10−2𝑛 =

1

102𝑛=

1

102

𝑛.

A.

Which will have the least value?

Step 1: To make comparison meaningful rewrite all expressions to have the same power

A.1

5

𝑛B. 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

𝑛The power is ‘n’. So, nothing needs to be done

B. 2-n Rule: 𝑎−𝑥 =1

𝑎𝑥. So, 2−𝑛 =

1

2𝑛=

1

2

𝑛

C. 10-2n Rule: 𝑎−𝑥 =1

𝑎𝑥. So, 10−2𝑛 =

1

102𝑛=

1

102

𝑛. Which is

1

100

𝑛

A.

Which will have the least value?

Step 1: To make comparison meaningful rewrite all expressions to have the same power

A.1

5

𝑛B. 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

𝑛The power is ‘n’. So, nothing needs to be done

B. 2-n Rule: 𝑎−𝑥 =1

𝑎𝑥. So, 2−𝑛 =

1

2𝑛=

1

2

𝑛

C. 10-2n Rule: 𝑎−𝑥 =1

𝑎𝑥. So, 10−2𝑛 =

1

102𝑛=

1

102

𝑛. Which is

1

100

𝑛

A.

D. 4𝑛

2

Which will have the least value?

Step 1: To make comparison meaningful rewrite all expressions to have the same power

A.1

5

𝑛B. 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

𝑛The power is ‘n’. So, nothing needs to be done

B. 2-n Rule: 𝑎−𝑥 =1

𝑎𝑥. So, 2−𝑛 =

1

2𝑛=

1

2

𝑛

C. 10-2n Rule: 𝑎−𝑥 =1

𝑎𝑥. So, 10−2𝑛 =

1

102𝑛=

1

102

𝑛. Which is

1

100

𝑛

A.

D. 4𝑛

2 Rule: 𝑎𝑥1

𝑦 = 𝑎𝑥

𝑦 =𝑦𝑎𝑥.

Which will have the least value?

Step 1: To make comparison meaningful rewrite all expressions to have the same power

A.1

5

𝑛B. 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

𝑛The power is ‘n’. So, nothing needs to be done

B. 2-n Rule: 𝑎−𝑥 =1

𝑎𝑥. So, 2−𝑛 =

1

2𝑛=

1

2

𝑛

C. 10-2n Rule: 𝑎−𝑥 =1

𝑎𝑥. So, 10−2𝑛 =

1

102𝑛=

1

102

𝑛. Which is

1

100

𝑛

A.

D. 4𝑛

2 Rule: 𝑎𝑥1

𝑦 = 𝑎𝑥

𝑦 =𝑦𝑎𝑥. So, 4

𝑛

2 =24

𝑛= 2𝑛

Which will have the least value?

Step 1: To make comparison meaningful rewrite all expressions to have the same power

A.1

5

𝑛B. 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

𝑛The power is ‘n’. So, nothing needs to be done

B. 2-n Rule: 𝑎−𝑥 =1

𝑎𝑥. So, 2−𝑛 =

1

2𝑛=

1

2

𝑛

C. 10-2n Rule: 𝑎−𝑥 =1

𝑎𝑥. So, 10−2𝑛 =

1

102𝑛=

1

102

𝑛. Which is

1

100

𝑛

A.

D. 4𝑛

2 Rule: 𝑎𝑥1

𝑦 = 𝑎𝑥

𝑦 =𝑦𝑎𝑥. So, 4

𝑛

2 =24

𝑛= 2𝑛

E. (0.05)-n

Which will have the least value?

Step 1: To make comparison meaningful rewrite all expressions to have the same power

A.1

5

𝑛B. 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

𝑛The power is ‘n’. So, nothing needs to be done

B. 2-n Rule: 𝑎−𝑥 =1

𝑎𝑥. So, 2−𝑛 =

1

2𝑛=

1

2

𝑛

C. 10-2n Rule: 𝑎−𝑥 =1

𝑎𝑥. So, 10−2𝑛 =

1

102𝑛=

1

102

𝑛. Which is

1

100

𝑛

A.

D. 4𝑛

2 Rule: 𝑎𝑥1

𝑦 = 𝑎𝑥

𝑦 =𝑦𝑎𝑥. So, 4

𝑛

2 =24

𝑛= 2𝑛

E. (0.05)-n = 5

100

−𝑛

Which will have the least value?

Step 1: To make comparison meaningful rewrite all expressions to have the same power

A.1

5

𝑛B. 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

𝑛The power is ‘n’. So, nothing needs to be done

B. 2-n Rule: 𝑎−𝑥 =1

𝑎𝑥. So, 2−𝑛 =

1

2𝑛=

1

2

𝑛

C. 10-2n Rule: 𝑎−𝑥 =1

𝑎𝑥. So, 10−2𝑛 =

1

102𝑛=

1

102

𝑛. Which is

1

100

𝑛

A.

D. 4𝑛

2 Rule: 𝑎𝑥1

𝑦 = 𝑎𝑥

𝑦 =𝑦𝑎𝑥. So, 4

𝑛

2 =24

𝑛= 2𝑛

E. (0.05)-n = 5

100

−𝑛=

1

20

−𝑛.

Which will have the least value?

Step 1: To make comparison meaningful rewrite all expressions to have the same power

A.1

5

𝑛B. 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

𝑛The power is ‘n’. So, nothing needs to be done

B. 2-n Rule: 𝑎−𝑥 =1

𝑎𝑥. So, 2−𝑛 =

1

2𝑛=

1

2

𝑛

C. 10-2n Rule: 𝑎−𝑥 =1

𝑎𝑥. So, 10−2𝑛 =

1

102𝑛=

1

102

𝑛. Which is

1

100

𝑛

A.

D. 4𝑛

2 Rule: 𝑎𝑥1

𝑦 = 𝑎𝑥

𝑦 =𝑦𝑎𝑥. So, 4

𝑛

2 =24

𝑛= 2𝑛

E. (0.05)-n = 5

100

−𝑛=

1

20

−𝑛. Rule: 𝑎−𝑥 =

1

𝑎𝑥.

Which will have the least value?

Step 1: To make comparison meaningful rewrite all expressions to have the same power

A.1

5

𝑛B. 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

𝑛The power is ‘n’. So, nothing needs to be done

B. 2-n Rule: 𝑎−𝑥 =1

𝑎𝑥. So, 2−𝑛 =

1

2𝑛=

1

2

𝑛

C. 10-2n Rule: 𝑎−𝑥 =1

𝑎𝑥. So, 10−2𝑛 =

1

102𝑛=

1

102

𝑛. Which is

1

100

𝑛

A.

D. 4𝑛

2 Rule: 𝑎𝑥1

𝑦 = 𝑎𝑥

𝑦 =𝑦𝑎𝑥. So, 4

𝑛

2 =24

𝑛= 2𝑛

E. (0.05)-n = 5

100

−𝑛=

1

20

−𝑛. Rule: 𝑎−𝑥 =

1

𝑎𝑥. So,

1

20

−𝑛= 20n

Which will have the least value?A.

1

5

𝑛B. 2-n C. 10-2n D. 4

𝑛

2 E. (0.05)-n

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

𝑛B. 2-n C. 10-2n D. 4

𝑛

2 E. (0.05)-n

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

𝑛B. 2-n C. 10-2n D. 4

𝑛

2 E. (0.05)-n

Listing down only the bases for all 5 choices

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

𝑛B. 2-n C. 10-2n D. 4

𝑛

2 E. (0.05)-n

Listing down only the bases for all 5 choices

A.1

5B.

1

2C.

1

100D. 2 E. 20

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

𝑛B. 2-n C. 10-2n D. 4

𝑛

2 E. (0.05)-n

Listing down only the bases for all 5 choices

A.1

5B.

1

2C.

1

100D. 2 E. 20

Of the 5 choices, the smallest number is 1

100

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

𝑛B. 2-n C. 10-2n D. 4

𝑛

2 E. (0.05)-n

Listing down only the bases for all 5 choices

A.1

5B.

1

2C.

1

100D. 2 E. 20

Of the 5 choices, the smallest number is 1

100

10-2n is the least value

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

𝑛B. 2-n C. 10-2n D. 4

𝑛

2 E. (0.05)-n

Listing down only the bases for all 5 choices

A.1

5B.

1

2C.

1

100D. 2 E. 20

Of the 5 choices, the smallest number is 1

100

Choices C is the answer.

10-2n is the least value

For more questions

Visit http://www.4gmat.com

GMAT Classes and GMAT Preparation

Send your comments / feedback to

info@4gmat.com

4GMATWe offer classroom training in Chennai and Bangalore

Tutors include GMAT 98%ilers, US B School graduates and IIM graduates

@ Chennai : +91 95000 48484

@ Bangalore: +91 74060 48484