GMAT Diagnostic Test Q2 - Problem Solving - Number Properties : Indices
Transcript of GMAT Diagnostic Test Q2 - Problem Solving - Number Properties : Indices
Question
For integer n > 1, which of the following expressions will have
the least value?
A.1
5
π
B. 2-n
C. 10-2n
D. 4π
2
E. (0.05)-n
Indices Rules
ππ₯ Γ ππ¦ = ππ₯+π¦ 102 Γ 103 = 105
Rule Example
ππ₯ π¦ = ππ₯π¦
Indices Rules
ππ₯ Γ ππ¦ = ππ₯+π¦ 102 Γ 103 = 105
Rule Example
ππ₯ π¦ = ππ₯π¦ 102 3 = 106
Indices Rules
ππ₯ Γ ππ¦ = ππ₯+π¦ 102 Γ 103 = 105
Rule Example
ππ₯ π¦ = ππ₯π¦ 102 3 = 106
ππ₯
ππ¦= ππ₯βπ¦
Indices Rules
ππ₯ Γ ππ¦ = ππ₯+π¦ 102 Γ 103 = 105
Rule Example
ππ₯ π¦ = ππ₯π¦ 102 3 = 106
ππ₯
ππ¦= ππ₯βπ¦
105
103= 102
Indices Rules
ππ₯ Γ ππ¦ = ππ₯+π¦ 102 Γ 103 = 105
Rule Example
ππ₯ π¦ = ππ₯π¦ 102 3 = 106
ππ₯
ππ¦= ππ₯βπ¦
105
103= 102
ππ₯1π¦ = π
π₯π¦ =
π¦ππ₯
Indices Rules
ππ₯ Γ ππ¦ = ππ₯+π¦ 102 Γ 103 = 105
Rule Example
ππ₯ π¦ = ππ₯π¦ 102 3 = 106
ππ₯
ππ¦= ππ₯βπ¦
105
103= 102
ππ₯1π¦ = π
π₯π¦ =
π¦ππ₯ 106
13 = 10
63 = 102
Indices Rules
ππ₯ Γ ππ¦ = ππ₯+π¦ 102 Γ 103 = 105
Rule Example
ππ₯ π¦ = ππ₯π¦ 102 3 = 106
ππ₯
ππ¦= ππ₯βπ¦
105
103= 102
ππ₯1π¦ = π
π₯π¦ =
π¦ππ₯ 106
13 = 10
63 = 102
10613 =
3106 = 102
4th rule can also be
expressed as
Indices Rules
ππ₯ Γ ππ¦ = ππ₯+π¦ 102 Γ 103 = 105
Rule Example
ππ₯ π¦ = ππ₯π¦ 102 3 = 106
ππ₯
ππ¦= ππ₯βπ¦
105
103= 102
ππ₯1π¦ = π
π₯π¦ =
π¦ππ₯
Rule Example
10613 = 10
63 = 102
10613 =
3106 = 102
4th rule can also be
expressed as
Indices Rules
ππ₯ Γ ππ¦ = ππ₯+π¦ 102 Γ 103 = 105
Rule Example
ππ₯ π¦ = ππ₯π¦ 102 3 = 106
ππ₯
ππ¦= ππ₯βπ¦
105
103= 102
ππ₯1π¦ = π
π₯π¦ =
π¦ππ₯
Rule Example
10613 = 10
63 = 102
πβπ₯ =1
ππ₯
10613 =
3106 = 102
4th rule can also be
expressed as
Indices Rules
ππ₯ Γ ππ¦ = ππ₯+π¦ 102 Γ 103 = 105
Rule Example
ππ₯ π¦ = ππ₯π¦ 102 3 = 106
ππ₯
ππ¦= ππ₯βπ¦
105
103= 102
ππ₯1π¦ = π
π₯π¦ =
π¦ππ₯
Rule Example
10613 = 10
63 = 102
πβπ₯ =1
ππ₯2β3 =
1
23
10613 =
3106 = 102
4th rule can also be
expressed as
Indices Rules
ππ₯ Γ ππ¦ = ππ₯+π¦ 102 Γ 103 = 105
Rule Example
ππ₯ π¦ = ππ₯π¦ 102 3 = 106
ππ₯
ππ¦= ππ₯βπ¦
105
103= 102
ππ₯1π¦ = π
π₯π¦ =
π¦ππ₯
Rule Example
10613 = 10
63 = 102
πβπ₯ =1
ππ₯2β3 =
1
23
ππ₯ Γ ππ₯ = ππ π₯
10613 =
3106 = 102
4th rule can also be
expressed as
Indices Rules
ππ₯ Γ ππ¦ = ππ₯+π¦ 102 Γ 103 = 105
Rule Example
ππ₯ π¦ = ππ₯π¦ 102 3 = 106
ππ₯
ππ¦= ππ₯βπ¦
105
103= 102
ππ₯1π¦ = π
π₯π¦ =
π¦ππ₯
Rule Example
10613 = 10
63 = 102
πβπ₯ =1
ππ₯2β3 =
1
23
ππ₯ Γ ππ₯ = ππ π₯ 23 Γ 33 = 63
10613 =
3106 = 102
4th rule can also be
expressed as
Which will have the least value?A.
1
5
πB. 2-n C. 10-2n D. 4
π
2 E. (0.05)-n
All of the answer choices have an βnβ 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
πB. 2-n C. 10-2n D. 4
π
2 E. (0.05)-n
All of the answer choices have an βnβ 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
πB. 2-n C. 10-2n D. 4
π
2 E. (0.05)-n
All of the answer choices have an βnβ term in their index.
1
5
πA.
Which will have the least value?
Step 1: To make comparison meaningful rewrite all expressions to have the same power
A.1
5
πB. 2-n C. 10-2n D. 4
π
2 E. (0.05)-n
All of the answer choices have an βnβ term in their index.
1
5
πThe power is βnβ. 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
πB. 2-n C. 10-2n D. 4
π
2 E. (0.05)-n
All of the answer choices have an βnβ term in their index.
1
5
πThe power is βnβ. 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
πB. 2-n C. 10-2n D. 4
π
2 E. (0.05)-n
All of the answer choices have an βnβ term in their index.
1
5
πThe power is βnβ. So, nothing needs to be done
B. 2-n Rule: πβπ₯ =1
ππ₯.
A.
Which will have the least value?
Step 1: To make comparison meaningful rewrite all expressions to have the same power
A.1
5
πB. 2-n C. 10-2n D. 4
π
2 E. (0.05)-n
All of the answer choices have an βnβ term in their index.
1
5
πThe power is βnβ. So, nothing needs to be done
B. 2-n Rule: πβπ₯ =1
ππ₯. So, 2βπ =
1
2π=
1
2
π
A.
Which will have the least value?
Step 1: To make comparison meaningful rewrite all expressions to have the same power
A.1
5
πB. 2-n C. 10-2n D. 4
π
2 E. (0.05)-n
All of the answer choices have an βnβ term in their index.
1
5
πThe power is βnβ. So, nothing needs to be done
B. 2-n Rule: πβπ₯ =1
ππ₯. So, 2βπ =
1
2π=
1
2
π
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
πB. 2-n C. 10-2n D. 4
π
2 E. (0.05)-n
All of the answer choices have an βnβ term in their index.
1
5
πThe power is βnβ. So, nothing needs to be done
B. 2-n Rule: πβπ₯ =1
ππ₯. So, 2βπ =
1
2π=
1
2
π
C. 10-2n Rule: πβπ₯ =1
ππ₯.
A.
Which will have the least value?
Step 1: To make comparison meaningful rewrite all expressions to have the same power
A.1
5
πB. 2-n C. 10-2n D. 4
π
2 E. (0.05)-n
All of the answer choices have an βnβ term in their index.
1
5
πThe power is βnβ. So, nothing needs to be done
B. 2-n Rule: πβπ₯ =1
ππ₯. So, 2βπ =
1
2π=
1
2
π
C. 10-2n Rule: πβπ₯ =1
ππ₯. So, 10β2π =
1
102π=
1
102
π.
A.
Which will have the least value?
Step 1: To make comparison meaningful rewrite all expressions to have the same power
A.1
5
πB. 2-n C. 10-2n D. 4
π
2 E. (0.05)-n
All of the answer choices have an βnβ term in their index.
1
5
πThe power is βnβ. So, nothing needs to be done
B. 2-n Rule: πβπ₯ =1
ππ₯. So, 2βπ =
1
2π=
1
2
π
C. 10-2n Rule: πβπ₯ =1
ππ₯. So, 10β2π =
1
102π=
1
102
π. Which is
1
100
π
A.
Which will have the least value?
Step 1: To make comparison meaningful rewrite all expressions to have the same power
A.1
5
πB. 2-n C. 10-2n D. 4
π
2 E. (0.05)-n
All of the answer choices have an βnβ term in their index.
1
5
πThe power is βnβ. So, nothing needs to be done
B. 2-n Rule: πβπ₯ =1
ππ₯. So, 2βπ =
1
2π=
1
2
π
C. 10-2n Rule: πβπ₯ =1
ππ₯. So, 10β2π =
1
102π=
1
102
π. Which is
1
100
π
A.
D. 4π
2
Which will have the least value?
Step 1: To make comparison meaningful rewrite all expressions to have the same power
A.1
5
πB. 2-n C. 10-2n D. 4
π
2 E. (0.05)-n
All of the answer choices have an βnβ term in their index.
1
5
πThe power is βnβ. So, nothing needs to be done
B. 2-n Rule: πβπ₯ =1
ππ₯. So, 2βπ =
1
2π=
1
2
π
C. 10-2n Rule: πβπ₯ =1
ππ₯. So, 10β2π =
1
102π=
1
102
π. Which is
1
100
π
A.
D. 4π
2 Rule: ππ₯1
π¦ = ππ₯
π¦ =π¦ππ₯.
Which will have the least value?
Step 1: To make comparison meaningful rewrite all expressions to have the same power
A.1
5
πB. 2-n C. 10-2n D. 4
π
2 E. (0.05)-n
All of the answer choices have an βnβ term in their index.
1
5
πThe power is βnβ. So, nothing needs to be done
B. 2-n Rule: πβπ₯ =1
ππ₯. So, 2βπ =
1
2π=
1
2
π
C. 10-2n Rule: πβπ₯ =1
ππ₯. So, 10β2π =
1
102π=
1
102
π. Which is
1
100
π
A.
D. 4π
2 Rule: ππ₯1
π¦ = ππ₯
π¦ =π¦ππ₯. So, 4
π
2 =24
π= 2π
Which will have the least value?
Step 1: To make comparison meaningful rewrite all expressions to have the same power
A.1
5
πB. 2-n C. 10-2n D. 4
π
2 E. (0.05)-n
All of the answer choices have an βnβ term in their index.
1
5
πThe power is βnβ. So, nothing needs to be done
B. 2-n Rule: πβπ₯ =1
ππ₯. So, 2βπ =
1
2π=
1
2
π
C. 10-2n Rule: πβπ₯ =1
ππ₯. So, 10β2π =
1
102π=
1
102
π. Which is
1
100
π
A.
D. 4π
2 Rule: ππ₯1
π¦ = ππ₯
π¦ =π¦ππ₯. So, 4
π
2 =24
π= 2π
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
πB. 2-n C. 10-2n D. 4
π
2 E. (0.05)-n
All of the answer choices have an βnβ term in their index.
1
5
πThe power is βnβ. So, nothing needs to be done
B. 2-n Rule: πβπ₯ =1
ππ₯. So, 2βπ =
1
2π=
1
2
π
C. 10-2n Rule: πβπ₯ =1
ππ₯. So, 10β2π =
1
102π=
1
102
π. Which is
1
100
π
A.
D. 4π
2 Rule: ππ₯1
π¦ = ππ₯
π¦ =π¦ππ₯. So, 4
π
2 =24
π= 2π
E. (0.05)-n = 5
100
βπ
Which will have the least value?
Step 1: To make comparison meaningful rewrite all expressions to have the same power
A.1
5
πB. 2-n C. 10-2n D. 4
π
2 E. (0.05)-n
All of the answer choices have an βnβ term in their index.
1
5
πThe power is βnβ. So, nothing needs to be done
B. 2-n Rule: πβπ₯ =1
ππ₯. So, 2βπ =
1
2π=
1
2
π
C. 10-2n Rule: πβπ₯ =1
ππ₯. So, 10β2π =
1
102π=
1
102
π. Which is
1
100
π
A.
D. 4π
2 Rule: ππ₯1
π¦ = ππ₯
π¦ =π¦ππ₯. So, 4
π
2 =24
π= 2π
E. (0.05)-n = 5
100
βπ=
1
20
βπ.
Which will have the least value?
Step 1: To make comparison meaningful rewrite all expressions to have the same power
A.1
5
πB. 2-n C. 10-2n D. 4
π
2 E. (0.05)-n
All of the answer choices have an βnβ term in their index.
1
5
πThe power is βnβ. So, nothing needs to be done
B. 2-n Rule: πβπ₯ =1
ππ₯. So, 2βπ =
1
2π=
1
2
π
C. 10-2n Rule: πβπ₯ =1
ππ₯. So, 10β2π =
1
102π=
1
102
π. Which is
1
100
π
A.
D. 4π
2 Rule: ππ₯1
π¦ = ππ₯
π¦ =π¦ππ₯. So, 4
π
2 =24
π= 2π
E. (0.05)-n = 5
100
βπ=
1
20
βπ. Rule: πβπ₯ =
1
ππ₯.
Which will have the least value?
Step 1: To make comparison meaningful rewrite all expressions to have the same power
A.1
5
πB. 2-n C. 10-2n D. 4
π
2 E. (0.05)-n
All of the answer choices have an βnβ term in their index.
1
5
πThe power is βnβ. So, nothing needs to be done
B. 2-n Rule: πβπ₯ =1
ππ₯. So, 2βπ =
1
2π=
1
2
π
C. 10-2n Rule: πβπ₯ =1
ππ₯. So, 10β2π =
1
102π=
1
102
π. Which is
1
100
π
A.
D. 4π
2 Rule: ππ₯1
π¦ = ππ₯
π¦ =π¦ππ₯. So, 4
π
2 =24
π= 2π
E. (0.05)-n = 5
100
βπ=
1
20
βπ. Rule: πβπ₯ =
1
ππ₯. So,
1
20
βπ= 20n
Step 2: Now that we have expressed all choices to power βnβ, just compare the bases
Which will have the least value?A.
1
5
πB. 2-n C. 10-2n D. 4
π
2 E. (0.05)-n
Step 2: Now that we have expressed all choices to power βnβ, just compare the bases
Which will have the least value?A.
1
5
πB. 2-n C. 10-2n D. 4
π
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 βnβ, just compare the bases
Which will have the least value?A.
1
5
πB. 2-n C. 10-2n D. 4
π
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 βnβ, just compare the bases
Which will have the least value?A.
1
5
πB. 2-n C. 10-2n D. 4
π
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 βnβ, just compare the bases
Which will have the least value?A.
1
5
πB. 2-n C. 10-2n D. 4
π
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 βnβ, just compare the bases
Which will have the least value?A.
1
5
πB. 2-n C. 10-2n D. 4
π
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
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