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If n = 10^10 and n^n = 10^d, what is the value of d? A. 10^

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by BTGmoderatorDC » Sun Aug 26, 2018 1:08 am
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If n = 10^10 and n^n = 10^d, what is the value of d?

A. 10^3
B. 10^10
C. 10^11
D. 10^20
E. 10^100

OA C

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by GMATGuruNY » Sun Aug 26, 2018 2:12 am
BTGmoderatorDC wrote:If n = 10^10 and n^n = 10^d, what is the value of d?

A. 10^3
B. 10^10
C. 10^11
D. 10^20
E. 10^100
Substituting n= 10¹� into n^n = 10^d, we get:
(10¹� )^(10¹�)= 10^d

When an exponent is raised to a power, we MULTIPLY the exponents.
Multiplying the exponents in blue, we get:
10^(10*10¹�) = 10^d
10^(10¹¹) = 10^d

Since each side has the same base, the exponent in blue must be equal to the exponent in red:
10¹¹ = d.

The correct answer is C.
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by Brent@GMATPrepNow » Sun Aug 26, 2018 4:58 am
BTGmoderatorDC wrote:If n = 10^10 and n^n = 10^d, what is the value of d?

A. 10^3
B. 10^10
C. 10^11
D. 10^20
E. 10^100
A slightly different approach....

Given: n = 10^10

Also given: n^n = 10^d
Replace n in the base with 10^10 to get: (10^10)^n = 10^d
Apply Power of a Power rule on left side to get: 10^10n = 10^d
So, we can conclude that 10n = d
Now replace the remaining n with 10^10 to get: 10(10^10) = d
This is the same as: (10^1)(10^10) = d
Apply Product rule to get: 10^11 = d

Answer: C

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by Scott@TargetTestPrep » Sat Apr 13, 2019 5:59 pm
BTGmoderatorDC wrote:If n = 10^10 and n^n = 10^d, what is the value of d?

A. 10^3
B. 10^10
C. 10^11
D. 10^20
E. 10^100

OA C

Source: Manhattan Prep
Since n = 10^10, then n^n = (10^10)^(10^10) = 10^(10 x 10^10) = 10^(10^11). So d = 10^11.

Answer: C

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