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605-655 Level|   Algebra|   Exponents|      
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russ9
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If n = 10^10 and n^n = 10^d, what is the value of d? Updated on: 14 Sep 2013, 02:57
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.
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C
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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

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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!
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C
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Bunuel
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If n = 10^10 and n^n = 10^d, what is the value of d? 22 Oct 2013, 10:23
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SaraLotfy
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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If n = 10^10 and n^n = 10^d, what is the value of d? 24 Feb 2014, 22:40
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SaraLotfy
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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If n = 10^10 and n^n = 10^d, what is the value of d? Updated on: 22 Feb 2021, 11:24
Originally posted by BrentGMATPrepNow on 26 Aug 2018, 05:58.
Last edited by BrentGMATPrepNow on 22 Feb 2021, 11:24, edited 1 time in total.
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russ9
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

Key property: \((b^x)^y = b^{xy}\)

If \(n = 10^{10}\) then \(n^n = (10^{10})^{(10^{10})} = 10^d\)

At this point, it helps to compare \((10^{10})^{(10^{10})}\) with \((b^x)^y\)

We can see that \(b = 10\), \(x = 10\) and \(y = 10^{10}\)

This means \(xy\) \(= (10)(10^{10})=(10^1)(10^{10})= 10^{11}\)

In other words: \((10^{10})^{(10^{10})} = 10^{(10^{11})} = 10^d\)

So, \(d=10^{11}\)

Answer: C

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If n = 10^10 and n^n = 10^d, what is the value of d? 13 Sep 2013, 20:47
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russ9
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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If n = 10^10 and n^n = 10^d, what is the value of d? 22 Oct 2013, 07:16
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Can someone propose another approach?
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SaraLotfy
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If n = 10^10 and n^n = 10^d, what is the value of d? 22 Oct 2013, 09:58
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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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If n = 10^10 and n^n = 10^d, what is the value of d? Updated on: 24 Apr 2014, 00:37
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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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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If n = 10^10 and n^n = 10^d, what is the value of d? 30 Jun 2015, 10:25
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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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If n = 10^10 and n^n = 10^d, what is the value of d? 18 Mar 2018, 07:37
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russ9
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

Show 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!

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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If n = 10^10 and n^n = 10^d, what is the value of d? 04 Sep 2018, 04:28
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russ9
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.

Answer: C
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If n = 10^10 and n^n = 10^d, what is the value of d? 12 Sep 2018, 11:27
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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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If n = 10^10 and n^n = 10^d, what is the value of d? 22 Sep 2020, 00:31
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n = \(10^{10}\)

Take 'log' on both the sides:

=> logn = log \(10^{10}\)

=> logn = 10 log 10 [log \((a)^b\) = b loga and log 10 = 1]

=> logn = 10

\(n^n\) = \(10^d\)

Take 'log' on both the sides:

=> log \(n^{n}\) = log \(10^{d}\)

=> nlogn = d log 10

=> nlogn = d

=> 10^{10} * 10 = d

=> d =10^{11}

Answer C
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If n = 10^10 and n^n = 10^d, what is the value of d? 14 Feb 2025, 05:12
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This question can be solved with basic exponents rules.

N^n = [10^10]^10^10

We know [A^m]^n = A ^ m*n and we know A^m*A^n = A^m+n

so 10^1 * 10*10 = 10 ^11

Now we have [10]^10^11 which is equal to 10^d

so d = 10^11 (C)
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