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If n = 10^10 and n^n = 10^d

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If n = 10^10 and n^n = 10^d

by Needgmat » Thu Aug 04, 2016 5:46 am
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

OAC

If n^n = 10^d, then 10^10^10^10

If I drop the base, then I will get 10^10^10.

So how come the answer is [spoiler]10^11[/spoiler]

Please explain.

Many thanks in advance.

Kavin
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by DavidG@VeritasPrep » Thu Aug 04, 2016 6:11 am
Needgmat 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

OAC

If n^n = 10^d, then 10^10^10^10

If I drop the base, then I will get 10^10^10.

So how come the answer is [spoiler]10^11[/spoiler]

Please explain.

Many thanks in advance.

Kavin
Think about it this way. If n = 10^10, then and n^n = (10^10)^(10^10.)

Consider a simple case. If we have (2^3)^4, we multiply the exponents (in red) to get 2^(3*4) = 2^12.

By the same logic (10^10)^(10^10) = 10^(10 * 10^10) = 10^(10^11)
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by GMATGuruNY » Thu Aug 04, 2016 6:45 am
Needgmat 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
Since n = 10¹�, n^n = (10¹�)^(10¹�).

Multiplying the exponents in blue, we get:
(10¹�)^(10¹�) = 10^(10*10¹�) = 10^(10¹¹).

Thus, d = 10¹¹.

The correct answer is C.
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by DavidG@VeritasPrep » Thu Aug 04, 2016 6:50 am
Needgmat 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

OAC

If n^n = 10^d, then 10^10^10^10

If I drop the base, then I will get 10^10^10.

So how come the answer is [spoiler]10^11[/spoiler]

Please explain.

Many thanks in advance.

Kavin
Alternatively, you could do the substitution in multiple steps. (substitutions in red)

If n = 10^10, then n^n = (10^10)^n
(10^10)^n = 10^(10 *n)
10^(10 *n) = 10^(10 * 10^10)
10^(10 * 10^10) = 10^(10^11)
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by Matt@VeritasPrep » Thu Aug 04, 2016 8:20 pm
It might be easiest to think of it this way:

(aᵇ)ᶜ = aᵇᶜ

In our case

(10¹�)^(10¹�) becomes

10^(10 * 10¹�)

or

10^(10¹ * 10¹�)

or

10^(10¹¹)
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