NEW HERE? LEARN MORE
closeclose
Last visit was: 23 Apr 2024, 19:14 It is currently 23 Apr 2024, 19:14
Decision Tracker
My Rewards
New posts
New comers' posts

Close
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Go to My Error Log Learn more
Close
Request Expert Reply
Confirm Cancel

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

SORT BY:
Date
Kudos
Tags:
Find Similar Topics 
Show Tags
Hide Tags
User avatar
russ9
Manager
Manager
Joined: 15 Aug 2013
Posts: 180
Own Kudos [?]: 330 [101]
Given Kudos: 23
Send PM
If n = 10^10 and n^n = 10^d, what is the value of d? [#permalink] New post   Updated on: 14 Sep 2013, 02:57
10
Kudos
91
Bookmarks
00:00
A
B
C
D
E

Difficulty:

55% (hard)

Question Stats:

58% (01:19) correct 42% (01:27) wrong based on 1125 sessions
Hide Show timer Statistics
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!
ShowHide Answer
Official Answer
C

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.
Most Helpful Reply
User avatar
L
Bunuel
Math Expert
Joined: 02 Sep 2009
Posts: 92883
Own Kudos [?]: 618606 [32]
Given Kudos: 81563
Send PM
CAT Tests Forum Quiz Mode
Re: If n = 10^10 and n^n = 10^d, what is the value of d? [#permalink] New post   22 Oct 2013, 10:23
13
Kudos
19
Bookmarks
Expert Reply
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.
User avatar
L
KarishmaB
Tutor
Joined: 16 Oct 2010
Posts: 14816
Own Kudos [?]: 64883 [15]
Given Kudos: 426
Location: Pune, India
Send PM
Re: If n = 10^10 and n^n = 10^d, what is the value of d? [#permalink] New post   24 Feb 2014, 22:40
13
Kudos
2
Bookmarks
Expert Reply
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 \(d = 100000000000 = 10^{11}\)
User avatar
L
BrentGMATPrepNow
GMAT Club Legend
GMAT Club Legend
Joined: 12 Sep 2015
Posts: 6821
Own Kudos [?]: 29894 [8]
Given Kudos: 799
Location: Canada
Send PM
If n = 10^10 and n^n = 10^d, what is the value of d? [#permalink] New post   Updated on: 22 Feb 2021, 11:24
7
Kudos
1
Bookmarks
Expert Reply
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


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

RELATED VIDEO

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.
General Discussion
User avatar
rrsnathan
Manager
Manager
Joined: 30 May 2013
Posts: 126
Own Kudos [?]: 358 [1]
Given Kudos: 72
Location: India
Concentration: Entrepreneurship, General Management
Schools: Booth '16 Carroll '16 ESADE '16 Foster '16 HEC Jan'16 Haas EWMBA '16 Tuck '16
GPA: 3.82
Send PM
GMAT ToolKit User
Re: If n = 10^10 and n^n = 10d, what is the value of d? [#permalink] New post   13 Sep 2013, 20:47
1
Kudos
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
avatar
waltiebikkiebal
Intern
Intern
Joined: 14 Feb 2012
Posts: 16
Own Kudos [?]: 17 [0]
Given Kudos: 6
Send PM
Re: If n = 10^10 and n^n = 10^d, what is the value of d? [#permalink] New post   22 Oct 2013, 07:16
Can someone propose another approach?
avatar
SaraLotfy
Intern
Intern
Joined: 24 Apr 2013
Posts: 44
Own Kudos [?]: 26 [1]
Given Kudos: 23
Location: United States
Send PM
Re: If n = 10^10 and n^n = 10^d, what is the value of d? [#permalink] New post   22 Oct 2013, 09:58
1
Bookmarks
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
avatar
PareshGmat
SVP
SVP
Joined: 27 Dec 2012
Status:The Best Or Nothing
Posts: 1562
Own Kudos [?]: 7207 [2]
Given Kudos: 193
Location: India
Concentration: General Management, Technology
WE:Information Technology (Computer Software)
Send PM
Re: If n = 10^10 and n^n = 10^d, what is the value of d? [#permalink] New post   Updated on: 24 Apr 2014, 00:37
1
Kudos
1
Bookmarks
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

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

\(d = 10^{11} = Answer = C\)

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.
User avatar
B
TimeTraveller
Senior Manager
Senior Manager
Joined: 28 Jun 2015
Posts: 250
Own Kudos [?]: 293 [0]
Given Kudos: 47
Concentration: Finance
GPA: 3.5
Send PM
Re: If n = 10^10 and n^n = 10^d, what is the value of d? [#permalink] New post   30 Jun 2015, 10:25
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
avatar
B
dineshril
Intern
Intern
Joined: 27 Apr 2015
Posts: 35
Own Kudos [?]: 47 [2]
Given Kudos: 238
GMAT 1: 370 Q29 V13
Send PM
Reviews Badge
Re: If n = 10^10 and n^n = 10^d, what is the value of d? [#permalink] New post   18 Mar 2018, 07:37
1
Kudos
1
Bookmarks
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

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
User avatar
P
JeffTargetTestPrep
Target Test Prep Representative
Joined: 04 Mar 2011
Status:Head GMAT Instructor
Affiliations: Target Test Prep
Posts: 3043
Own Kudos [?]: 6271 [0]
Given Kudos: 1646
Send PM
Re: If n = 10^10 and n^n = 10^d, what is the value of d? [#permalink] New post   04 Sep 2018, 04:28
Expert Reply
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.

Answer: C
User avatar
S
arpitkansal
Intern
Intern
Joined: 17 Jun 2018
Posts: 44
Own Kudos [?]: 39 [0]
Given Kudos: 478
Location: Canada
Schools: IMD '20
GMAT 1: 690 Q48 V36
GPA: 2.84
WE:Engineering (Real Estate)
Send PM
Re: If n = 10^10 and n^n = 10^d, what is the value of d? [#permalink] New post   12 Sep 2018, 11:27
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
User avatar
L
MathRevolution
Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 10161
Own Kudos [?]: 16592 [0]
Given Kudos: 4
GMAT 1: 760 Q51 V42
GPA: 3.82
Send PM
Re: If n = 10^10 and n^n = 10^d, what is the value of d? [#permalink] New post   22 Sep 2020, 00:31
Expert Reply
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
User avatar
G
Basshead
Director
Director
Joined: 09 Jan 2020
Posts: 967
Own Kudos [?]: 223 [0]
Given Kudos: 434
Location: United States
Send PM
Re: If n = 10^10 and n^n = 10^d, what is the value of d? [#permalink] New post   24 Oct 2020, 14:48
BrentGMATPrepNow wrote:
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

Answer: C

RELATED VIDEO FROM OUR COURSE


Great approach.

It was confusing to grasp the concept, but your explanation made everything very clear. Kudos.
User avatar
L
BrentGMATPrepNow
GMAT Club Legend
GMAT Club Legend
Joined: 12 Sep 2015
Posts: 6821
Own Kudos [?]: 29894 [0]
Given Kudos: 799
Location: Canada
Send PM
Re: If n = 10^10 and n^n = 10^d, what is the value of d? [#permalink] New post   10 Mar 2022, 13:26
Expert Reply
Top Contributor
Basshead wrote:
Great approach.
It was confusing to grasp the concept, but your explanation made everything very clear. Kudos.


Thanks! I'm glad you liked it!
User avatar
bumpbot
Non-Human User
Joined: 09 Sep 2013
Posts: 32630
Own Kudos [?]: 821 [0]
Given Kudos: 0
Send PM
Re: If n = 10^10 and n^n = 10^d, what is the value of d? [#permalink] New post   23 Apr 2024, 06:44
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
GMAT Club Bot
gmatclubot
Re: If n = 10^10 and n^n = 10^d, what is the value of d? [#permalink]
23 Apr 2024, 06:44
Moderators:
Bunuel
Math Expert
92883 posts
yashikaaggarwal
Senior Moderator - Masters Forum
3137 posts

Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne
Main navigation
GMAT Resources
Partners
MBA Resources
www.gmac.com|www.mba.com | | GMAT Club Rules | Terms and Conditions