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Manager  Joined: 15 Aug 2013
Posts: 228
If n = 10^10 and n^n = 10^d, what is the value of d?  [#permalink]

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Difficulty:   55% (hard)

Question Stats: 56% (01:18) correct 44% (01:28) wrong based on 583 sessions

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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

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!

Originally posted by russ9 on 13 Sep 2013, 17:36.
Last edited by Bunuel on 14 Sep 2013, 02:57, edited 3 times in total.
Edited the question.
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Re: If n = 10^10 and n^n = 10^d, what is the value of d?  [#permalink]

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7
11
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 $$n^n=(10^{10})^{(10^{10})}=10^{10*10^{10}}=10^{10^{11}}$$.

$$10^{10^{11}}=10^d$$ --> $$d=10^{11}$$.

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Re: If n = 10^10 and n^n = 10d, what is the value of d?  [#permalink]

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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

Regards,
Rrsnathan
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Re: If n = 10^10 and n^n = 10^d, what is the value of d?  [#permalink]

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Can someone propose another approach?
Manager  Joined: 24 Apr 2013
Posts: 51
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Re: If n = 10^10 and n^n = 10^d, what is the value of d?  [#permalink]

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1
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]

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1
1
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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Kindly press "+1 Kudos" to appreciate Originally posted by PareshGmat on 24 Feb 2014, 03:18.
Last edited by PareshGmat on 24 Apr 2014, 00:37, 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]

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2
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]

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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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GMAT 1: 370 Q29 V13 Re: If n = 10^10 and n^n = 10^d, what is the value of d?  [#permalink]

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1
russ9 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

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!

Given
=> $$n = 10^{10}$$ -----(1)
=> $$n^n = 10^d$$------(2)

Now from (2) we have
=> $$n^n = 10^d$$
=> OR $$n = 10^{\frac{d}{n}}$$
=> OR $$10^{10} = 10^{\frac{d}{n}}$$---- substituting 'n' from (1) in LHS

Since base is same comparing both the sides we have
=> $$10=\frac{d}{n}$$

=> OR $$d=10*n$$
=> OR $$d=10*10^{10}$$ ------ substituting 'n' from (1)

=> OR $$d=10^{1+10}=10^{11}$$

Option "C"

Thanks
Dinesh
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Re: If n = 10^10 and n^n = 10^d, what is the value of d?  [#permalink]

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Top Contributor
russ9 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

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Re: If n = 10^10 and n^n = 10^d, what is the value of d?  [#permalink]

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russ9 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 = 10^10, n^n = (10^10)^(10^10) = 10^(10 x 10^10) = 10^(10^11). So d = 10^11.

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Re: If n = 10^10 and n^n = 10^d, what is the value of d?  [#permalink]

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We can solve this by logarithm to the base 10 approach also.
we know that log 10 = 1
n=10^10
this implies that log n=10

n^n=10^d

taking log to the base 10 gives
n log n = d

so 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]

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