NEW HERE? LEARN MORE
closeclose
It is currently 25 Sep 2023, 03:08
Register
GMAT Club Tests
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?

16 posts Go to First Unread Post
Print view
NEW TOPIC
POST REPLY
Downloads
My Bookmarks
Reviews
SORT BY:
Date
Kudos
User avatar
russ9
Manager
Manager
Joined: 15 Aug 2013
Posts: 184
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
9
Kudos
85
Bookmarks
Quiz Dashboard Attempt only this question Same topic and source Same topic and difficulty Same source and difficulty Random Chronological Reverse chronological Learn more
00:00
A
B
C
D
E

Difficulty:

(N/A)

Question Stats:

57% (01:18) correct 43% (01:27) wrong based on 1062 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
Official Answer and Stats are available only to registered users. Register/Login.

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.
Quiz Dashboard Attempt only this question Same topic and source Same topic and difficulty Same source and difficulty Random Chronological Reverse chronological Learn more
Most Helpful Reply
User avatar
L
Bunuel
Math Expert
Joined: 02 Sep 2009
Posts: 89522
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
12
Kudos
18
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.
_________________
New to the GMAT CLUB Forum?

My Signature Questions' Collection:


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics
Signature Read More
User avatar
L
KarishmaB
Tutor
Joined: 16 Oct 2010
Posts: 14149
Location: Pune, India
Re: If n = 10^10 and n^n = 10^d, what is the value of d? [#permalink] New post  24 Feb 2014, 22:40
12
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 \(d = 100000000000 = 10^{11}\)
_________________
Karishma Bansal - ANA PREP

Register for FREE trial of our complete GMAT Prep Package at ANA PREP
(Includes trial access to Study Modules, Conceptual Videos, Practice Questions and LIVE Webinars)

YouTube Channel @karishma.anaprep
youtube.com/channel/UCVxczpVAbQSQGEUyt72G2Eg/
Signature Read More
User avatar
L
BrentGMATPrepNow
GMAT Club Legend
GMAT Club Legend
Joined: 12 Sep 2015
Posts: 6881
Location: Canada
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
5
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: 129
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
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
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
Location: United States
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
User avatar
PareshGmat
SVP
SVP
Joined: 27 Dec 2012
Status:The Best Or Nothing
Posts: 1573
Location: India
Concentration: General Management, Technology
WE:Information Technology (Computer Software)
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: 253
Concentration: Finance
GPA: 3.5
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
_________________
I used to think the brain was the most important organ. Then I thought, look what’s telling me that.
Signature Read More
avatar
B
dineshril
Intern
Intern
Joined: 27 Apr 2015
Posts: 35
GMAT 1: 370 Q29 V13
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
G
JeffTargetTestPrep
Target Test Prep Representative
Joined: 04 Mar 2011
Status:Head GMAT Instructor
Affiliations: Target Test Prep
Posts: 3013
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
_________________
Signature Read More
User avatar
S
arpitkansal
Intern
Intern
Joined: 17 Jun 2018
Posts: 46
Location: Canada
Schools: IMD '20
GMAT 1: 690 Q48 V36
GPA: 2.84
WE:Engineering (Real Estate)
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: 10461
GMAT 1: 760 Q51 V42
GPA: 3.82
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
P
Basshead
VP
VP
Joined: 09 Jan 2020
Posts: 1007
Location: United States
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.
_________________
Help me get better -- please critique my responses!
Signature Read More
User avatar
L
BrentGMATPrepNow
GMAT Club Legend
GMAT Club Legend
Joined: 12 Sep 2015
Posts: 6881
Location: Canada
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: 29472
Re: If n = 10^10 and n^n = 10^d, what is the value of d? [#permalink] New post  24 Mar 2023, 11:45
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 Books | GMAT Club Tests | Best Prices on GMAT Courses | GMAT Mobile App | Math Resources | Verbal Resources
Signature Read More
GMAT Club Bot
gmatclubot
Re: If n = 10^10 and n^n = 10^d, what is the value of d? [#permalink]
24 Mar 2023, 11:45
NEW TOPIC
POST REPLY
Downloads
My Bookmarks
Reviews
Moderators:
Bunuel
Math Expert
89522 posts
yashikaaggarwal
Senior Moderator - Masters Forum
3134 posts

cron

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