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

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If n = 10^10 and n^n = 10^d, what is the value of d? [#permalink] New post 13 Sep 2013, 16:36
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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

[Reveal] Spoiler:
I chose A and it was obviously incorrect but I can't figure out why. I thought that I could multiply exponents, doesn't that mean that I would essentially have, -(10^10)^(10^10) = 10^d
-10^10.10.10 = 10^d
-10^1000 = 10^d
d= 1,000 = 10^3?

Why doesn't that work?

Thanks!
[Reveal] Spoiler: OA

Last edited by Bunuel on 14 Sep 2013, 01:57, edited 3 times in total.
Edited the question.
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Re: If n = 10^10 and n^n = 10d, what is the value of d? [#permalink] New post 13 Sep 2013, 19:47
russ9 wrote:
If n = 10^10 and n^n = 10d, what is the value of d?


10^3
10^10
10^11
10^20
10^100

I chose A and it was obviously incorrect but I can't figure out why. I thought that I could multiply exponents, doesn't that mean that I would essentially have, -(10^10)^(10^10) = 10^d
-10^10.10.10 = 10^d
-10^1000 = 10^d
d= 1,000 = 10^3?

Why doesn't that work?

Thanks!



Hi,

n^n=
=(10^10)^(10^10)
this can be written as
= 10^(10*10^10)
=10^(10^11)

Refer Gmat Math book:

math-number-theory-88376.html#p666609

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Rrsnathan
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Re: If n = 10^10 and n^n = 10^d, what is the value of d? [#permalink] New post 22 Oct 2013, 06:16
Can someone propose another approach?
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Re: If n = 10^10 and n^n = 10^d, what is the value of d? [#permalink] New post 22 Oct 2013, 08:58
I know we should apply this rule:

\((a^m)^n\)=\(a^{mn}\)

a\(^m^n\)=\(a^{(m^n)}\)and not \((a^m)^n\)

but I still have a problem in applying it to this question
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Re: If n = 10^10 and n^n = 10^d, what is the value of d? [#permalink] New post 22 Oct 2013, 09:23
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SaraLotfy wrote:
I know we should apply this rule:

\((a^m)^n\)=\(a^{mn}\)

a\(^m^n\)=\(a^{(m^n)}\)and not \((a^m)^n\)

but I still have a problem in applying it to this question


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 \(n = 10^{10}\), then \(n^n=(10^{10})^{(10^{10})}=10^{10*10^{10}}=10^{10^{11}}\).

\(10^{10^{11}}=10^d\) --> \(d=10^{11}\).

Answer: C.
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Re: If n = 10^10 and n^n = 10^d, what is the value of d? [#permalink] New post 24 Feb 2014, 02:18
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Did in this way:

\(n = 10^{10}\)

\((10^{10})^{(10^{10})} = 10^d\) ........... As per equation

Can be re-written as

\((10^{10})^{(10^{10})} = 10^{(d * 10 * \frac{1}{10})}\)

can be re-written as

\((10^{10})^{(10^{10})} = (10^{10})^{(\frac{d}{10})}\)

Bases are same, so equating powers

\(\frac{d}{10} = 10^{10}\)

\(d = 10^{11} = Answer = C\)
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Last edited by PareshGmat on 23 Apr 2014, 23:37, edited 1 time in total.
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Re: If n = 10^10 and n^n = 10^d, what is the value of d? [#permalink] New post 24 Feb 2014, 21:40
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SaraLotfy wrote:
I know we should apply this rule:

\((a^m)^n\)=\(a^{mn}\)

a\(^m^n\)=\(a^{(m^n)}\)and not \((a^m)^n\)

but I still have a problem in applying it to this question


Yes, I guess many people will have a similar feeling. Think of it not in terms of exponents but simple numbers.

Imagine what 10^10 is: n = 10000000000 (it has 10 zeroes)
\(n^n = 10000000000^{10000000000} = (10^{10})^{10000000000} = 10^{100000000000}\) (now it has 11 zeroes)
\(10^d = 10^{100000000000}\)

So \(d = 100000000000 = 10^{11}\)
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Re: If n = 10^10 and n^n = 10^d, what is the value of d? [#permalink] New post 30 Jun 2015, 08:32
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Re: If n = 10^10 and n^n = 10^d, what is the value of d? [#permalink] New post 30 Jun 2015, 09:25
n = 10^10
n^n = 10^d
now, n^n = 10^10^n = 10^10n.
so, 10^d = 10^10n
d = 10(10^10) = 10^11
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Re: If n = 10^10 and n^n = 10^d, what is the value of d?   [#permalink] 30 Jun 2015, 09:25
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