Last visit was: 19 Nov 2025, 01:11 It is currently 19 Nov 2025, 01:11
Home
GMAT
Messages
Tests
Schools
Events
Rewards
Chat
My Profile
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
Back to Forum
Create Topic
555-605 Level|   Algebra|   Exponents|      
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 19 Nov 2025
Posts: 105,379
Own Kudos:
778,163
 [18]
Given Kudos: 99,977
Products:
GMAT Club Tests Forum Quiz
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 105,379
Kudos: 778,163
 [18]
If (3 + 4)(3^2 + 4^2)(3^4 + 4^4)(3^8 + 4^8)(3^16 + 4^16)(3^32 + 4^32) 16 Jun 2020, 06:58
Kudos
Add Kudos
18
Bookmarks
Bookmark this Post
00:00
Start Timer
Pause Timer
Resume Timer
Show Answer
Hide
Show
History
E
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!

Difficulty:

35% (medium)

Question Stats:

69% (01:49) correct 31% (02:22) wrong based on 170 sessions

History

Date
Time
Result
Not Attempted Yet
If \((3 + 4)(3^2 + 4^2)(3^4 + 4^4)(3^8 + 4^8)(3^{16} + 4^{16})(3^{32} + 4^{32}) = 4^x - 3^x\), what is the value of x?

A. 4
B. 8
C. 16
D. 32
E. 64


Are You Up For the Challenge: 700 Level Questions­
ShowHide Answer
Official Answer
E
User avatar
GMATinsight
User avatar
Major Poster
Joined: 08 Jul 2010
Last visit: 18 Nov 2025
Posts: 6,836
Own Kudos:
16,349
 [1]
Given Kudos: 128
Status:GMAT/GRE Tutor l Admission Consultant l On-Demand Course creator
Location: India
GMAT: QUANT+DI EXPERT
Schools: IIM (A) ISB '24
GMAT 1: 750 Q51 V41
WE:Education (Education)
Products:
My Reviews
Expert
Expert reply
Schools: IIM (A) ISB '24
GMAT 1: 750 Q51 V41
Posts: 6,836
Kudos: 16,349
 [1]
If (3 + 4)(3^2 + 4^2)(3^4 + 4^4)(3^8 + 4^8)(3^16 + 4^16)(3^32 + 4^32) 16 Jun 2020, 07:09
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Bunuel
If \((3 + 4)(3^2 + 4^2)(3^4 + 4^4)(3^8 + 4^8)(3^{16} + 4^{16})(3^{32} + 4^{32}) = 4^x - 3^x\), what is the value of x?

A. 4
B. 8
C. 16
D. 32
E. 64


Are You Up For the Challenge: 700 Level Questions
Show more
For x = 2, \(4^x - 3^x = 16 - 9 = 7\) which is equal to \(4+3\)
For x = 4, \(4^x - 3^x = 256 - 81 = 175\) which is equal to \((4+3)(4^2+3^2)\)

OBSERVATION: Value of x is Double the power of 4 and 3 in biggest expression \((4^2+3^2)\)

i,e, \((3 + 4)(3^2 + 4^2)(3^4 + 4^4)(3^8 + 4^8)(3^{16} + 4^{16})(3^{32} + 4^{32}) = 4^x - 3^x\)

i.e. x = 2*32 = 64



Answer: Option E­
User avatar
Lipun
User avatar
Joined: 05 Jan 2020
Last visit: 08 Jan 2025
Posts: 144
Own Kudos:
157
 [3]
Given Kudos: 291
Posts: 144
Kudos: 157
 [3]
If (3 + 4)(3^2 + 4^2)(3^4 + 4^4)(3^8 + 4^8)(3^16 + 4^16)(3^32 + 4^32) 16 Jun 2020, 07:41
2
Kudos
Add Kudos
1
Bookmarks
Bookmark this Post
(3 + 4)(3^2 + 4^2)(3^4 + 4^4)(3^8 + 4^8)(3^16 + 4^16)(3^32 + 4^32)

= (4-3)(3+4)(3^2 + 4^2)(3^4 + 4^4)(3^8 + 4^8)(3^16 + 4^16)(3^32 + 4^32)

=(4^2 - 3^2)(3^2 + 4^2)(3^4 + 4^4)(3^8 + 4^8)(3^16 + 4^16)(3^32 + 4^32)
.
. continue applying a^2 - b^2 = (a + b)(a - b)...after first two terms you can see that the successive term is twice in power
.
=4^64 - 3^64

So, x = 64
User avatar
HoneyLemon
User avatar
Stern School Moderator
Joined: 26 May 2020
Last visit: 02 Oct 2023
Posts: 628
Own Kudos:
565
 [1]
Given Kudos: 219
Status:Spirited
Concentration: General Management, Technology
Schools: Stern '23 LBS '23 Said '23 IIMB EPGP'22 INSEAD Jan'23 intake
WE:Analyst (Computer Software)
Schools: Stern '23 LBS '23 Said '23 IIMB EPGP'22 INSEAD Jan'23 intake
Posts: 628
Kudos: 565
 [1]
If (3 + 4)(3^2 + 4^2)(3^4 + 4^4)(3^8 + 4^8)(3^16 + 4^16)(3^32 + 4^32) 16 Jun 2020, 22:00
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Bunuel
If \((3 + 4)(3^2 + 4^2)(3^4 + 4^4)(3^8 + 4^8)(3^{16} + 4^{16})(3^{32} + 4^{32}) = 4^x - 3^x\), what is the value of x?

A. 4
B. 8
C. 16
D. 32
E. 64
Show more

Lets find the value for the smaller terms in the sequence .

----- ----------- Term-------------------------------Value--------------------------X = ?-----------

------ 3+4--------------------------------------7 = \(4 ^2-3^2\) --------------------X = 2-----------

----(3+4)(3^2+4^2)-----------------------175 = 256-81=\(4 ^4-3^4\) ----------X = 4-----------


So we can apply this knowledge in for
\((3 + 4)(3^2 + 4^2)(3^4 + 4^4)(3^8 + 4^8)(3^{16} + 4^{16})(3^{32} + 4^{32})\) = \( 4^x - 3^x\)
X must be 2 * 32 = 64 .

Ans is E
User avatar
chetan2u
User avatar
GMAT Expert
Joined: 02 Aug 2009
Last visit: 15 Nov 2025
Posts: 11,238
Own Kudos:
43,699
 [5]
Given Kudos: 335
Status:Math and DI Expert
Location: India
Concentration: Human Resources, General Management
GMAT Focus 1: 735 Q90 V89 DI81
Products:
My Reviews
Expert
Expert reply
GMAT Focus 1: 735 Q90 V89 DI81
Posts: 11,238
Kudos: 43,699
 [5]
If (3 + 4)(3^2 + 4^2)(3^4 + 4^4)(3^8 + 4^8)(3^16 + 4^16)(3^32 + 4^32) 16 Jun 2020, 23:45
4
Kudos
Add Kudos
1
Bookmarks
Bookmark this Post
Bunuel
If \((3 + 4)(3^2 + 4^2)(3^4 + 4^4)(3^8 + 4^8)(3^{16} + 4^{16})(3^{32} + 4^{32}) = 4^x - 3^x\), what is the value of x?

A. 4
B. 8
C. 16
D. 32
E. 64


Are You Up For the Challenge: 700 Level Questions
Show more

Algebraic method


The moment you see something of the form \(a^2+b^2\) and \(a^x-b^x\), try to use the formula => \(a^2-b^2=(a-b)(a+b)\).

Now \(a+b=\frac{a^2-b^2}{a-b}........a^2+b^2=\frac{a^4-b^4}{a^2-b^2}.......... a^4+b^4=\frac{a^8-b^8}{a^4-b^4}........a^{32}+b^{32}=\frac{a^{64}-b^{64}}{a^{32}-b^{32}}..........\)

\((3 + 4)(3^2 + 4^2)(3^4 + 4^4)(3^8 + 4^8)(3^{16} + 4^{16})(3^{32} + 4^{32})\)

\(=\frac{4^2-3^2}{4-3}*\frac{4^4-3^4}{4^2-3^2}*\frac{4^8-3^8}{4^4-3^4}*........*\frac{4^{64}-3^{64}}{4^{32}-3^{32}}\)

\(=\frac{4^{64}-3^{64}}{4-3}=4^{64}-3^{64}= 4^x - 3^x\)..

Thus, \(x=64\)

Choices


Now each term in \((3 + 4)(3^2 + 4^2)(3^4 + 4^4)(3^8 + 4^8)(3^{16} + 4^{16})(3^{32} + 4^{32})\) is positive and will surely be greater than \(3^{32} + 4^{32}\)
and \(3^{32} + 4^{32}\)>\(4^{32} -3^{32}\), so no choice \(\leq{32}\) can fit in.
\(x=64\)

E­
avatar
ORR1013
avatar
Joined: 14 Aug 2019
Last visit: 11 Jul 2020
Posts: 1
Given Kudos: 18
Posts: 1
Kudos: 0
If (3 + 4)(3^2 + 4^2)(3^4 + 4^4)(3^8 + 4^8)(3^16 + 4^16)(3^32 + 4^32) 01 Jul 2020, 18:42
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Lipun
(3 + 4)(3^2 + 4^2)(3^4 + 4^4)(3^8 + 4^8)(3^16 + 4^16)(3^32 + 4^32)

= (4-3)(3+4)(3^2 + 4^2)(3^4 + 4^4)(3^8 + 4^8)(3^16 + 4^16)(3^32 + 4^32)

=(4^2 - 3^2)(3^2 + 4^2)(3^4 + 4^4)(3^8 + 4^8)(3^16 + 4^16)(3^32 + 4^32)
.
. continue applying a^2 - b^2 = (a + b)(a - b)...after first two terms you can see that the successive term is twice in power
.
=4^64 - 3^64

So, x = 64
Show more

Bunuel

I don't quite follow how we can just pull a 4-3 out of the air and tack it on to the left. Could use a bit of help understanding this problem as the answers make a few leaps that make me confused. Thanks in advance!
User avatar
chetan2u
User avatar
GMAT Expert
Joined: 02 Aug 2009
Last visit: 15 Nov 2025
Posts: 11,238
Own Kudos:
43,699
 [1]
Given Kudos: 335
Status:Math and DI Expert
Location: India
Concentration: Human Resources, General Management
GMAT Focus 1: 735 Q90 V89 DI81
Products:
My Reviews
Expert
Expert reply
GMAT Focus 1: 735 Q90 V89 DI81
Posts: 11,238
Kudos: 43,699
 [1]
If (3 + 4)(3^2 + 4^2)(3^4 + 4^4)(3^8 + 4^8)(3^16 + 4^16)(3^32 + 4^32) 01 Jul 2020, 18:55
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
ORR1013
Lipun
(3 + 4)(3^2 + 4^2)(3^4 + 4^4)(3^8 + 4^8)(3^16 + 4^16)(3^32 + 4^32)

= (4-3)(3+4)(3^2 + 4^2)(3^4 + 4^4)(3^8 + 4^8)(3^16 + 4^16)(3^32 + 4^32)

=(4^2 - 3^2)(3^2 + 4^2)(3^4 + 4^4)(3^8 + 4^8)(3^16 + 4^16)(3^32 + 4^32)
.
. continue applying a^2 - b^2 = (a + b)(a - b)...after first two terms you can see that the successive term is twice in power
.
=4^64 - 3^64

So, x = 64

Bunuel

I don't quite follow how we can just pull a 4-3 out of the air and tack it on to the left. Could use a bit of help understanding this problem as the answers make a few leaps that make me confused. Thanks in advance!
Show more

4-3 is nothing but 1, so We can write Anything that adds up to 1 to a product and it will not make any change.
If we have x(x-7), we can write it as (7-6)(0.5+0.5)x(x-7).
Why? Because 7-6 is also 1 and 0.5+0.5 is also 1.

Here 4-3 is taken to relate it to 4+3 as (a-b)(a+b)=a^2-b^2
User avatar
bumpbot
User avatar
Non-Human User
Joined: 09 Sep 2013
Last visit: 04 Jan 2021
Posts: 38,584
Own Kudos:
1,079
Posts: 38,584
Kudos: 1,079
If (3 + 4)(3^2 + 4^2)(3^4 + 4^4)(3^8 + 4^8)(3^16 + 4^16)(3^32 + 4^32) 20 Jul 2024, 20:27
Kudos
Add Kudos
Bookmarks
Bookmark this Post
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.
Moderators:
Bunuel
Math Expert
105379 posts
Krunaal
Tuck School Moderator
805 posts