NEW!!! Tough and tricky exponents and roots questions

Sort by Date

Sort by Kudos

Author

Message

TAGS:

Hide Tags

Math Expert

Joined: 02 Sep 2009

Posts: 44599

NEW!!! Tough and tricky exponents and roots questions [#permalink]

Show Tags

12 Jan 2012, 03:03

This post received
KUDOS

Expert's post

328

This post was
BOOKMARKED

Exponents and roots problems are very common on the GMAT. So, it's extremely important to know how to manipulate them, how to factor out, take roots, multiply, divide, etc. Below are 11 problems to test your skills. Please post your thought process/solutions along with the answers.

I'll post OA's with detailed solutions tomorrow. Good luck.

1. What is the value of \(\sqrt{25+10\sqrt{6}}+\sqrt{25-10\sqrt{6}}\)?
A. \(2\sqrt{5}\)
B. \(\sqrt{55}\)
C. \(2\sqrt{15}\)
D. 50
E. 60

Solution: tough-and-tricky-exponents-and-roots-questions-125956-40.html#p1029216

2. What is the units digit of \((17^3)^4-1973^{3^2}\)?
A. 0
B. 2
C. 4
D. 6
E. 8

Solution: tough-and-tricky-exponents-and-roots-questions-125956-40.html#p1029219

3. If \(5^{10x}=4,900\) and \(2^{\sqrt{y}}=25\) what is the value of \(\frac{(5^{(x-1)})^5}{4^{-\sqrt{y}}}\)?
A. 14/5
B. 5
C. 28/5
D. 13
E. 14

Solution: tough-and-tricky-exponents-and-roots-questions-125956-40.html#p1029221

4. What is the value of \(5+4*5+4*5^2+4*5^3+4*5^4+4*5^5\)?
A. 5^6
B. 5^7
C. 5^8
D. 5^9
E. 5^10

Solution: tough-and-tricky-exponents-and-roots-questions-125956-40.html#p1029222

5. If \(x=23^2*25^4*27^6*29^8\) and is a multiple of \(26^n\), where \(n\) is a non-negative integer, then what is the value of \(n^{26}-26^n\)?
A. -26
B. -25
C. -1
D. 0
E. 1

Solution: tough-and-tricky-exponents-and-roots-questions-125956-40.html#p1029223

6. If \(x=\sqrt[5]{-37}\) then which of the following must be true?
A. \(\sqrt{-x}>2\)
B. x>-2
C. x^2<4
D. x^3<-8
E. x^4>32

Solution: tough-and-tricky-exponents-and-roots-questions-125956-40.html#p1029224

7. If \(x=\sqrt{10}+\sqrt[3]{9}+\sqrt[4]{8}+\sqrt[5]{7}+\sqrt[6]{6}+\sqrt[7]{5}+\sqrt[8]{4}+\sqrt[9]{3}+\sqrt[10]{2}\), then which of the following must be true:
A. x<6
B. 6<x<8
C. 8<x<10
D. 10<x<12
E. x>12

Solution: tough-and-tricky-exponents-and-roots-questions-125956-40.html#p1029227

8. If \(x\) is a positive number and equals to \(\sqrt{6+{\sqrt{6+\sqrt{6+\sqrt{6+...}}}}}\), where the given expression extends to an infinite number of roots, then what is the value of x?
A. \(\sqrt{6}\)
B. 3
C. \(1+\sqrt{6}\)
D. \(2\sqrt{3}\)
E. 6

Solution: tough-and-tricky-exponents-and-roots-questions-125956-40.html#p1029228

9. If \(x\) is a positive integer then the value of \(\frac{22^{22x}-22^{2x}}{11^{11x}-11^x}\) is closest to which of the following?
A. \(2^{11x}\)
B. \(11^{11x}\)
C. \(22^{11x}\)
D. \(2^{22x}*11^{11x}\)
E. \(2^{22x}*11^{22x}\)

Solution: tough-and-tricky-exponents-and-roots-questions-125956-40.html#p1029229

10. Given that \(5x=125-3y+z\) and \(\sqrt{5x}-5-\sqrt{z-3y}=0\), then what is the value of \(\sqrt{\frac{45(z-3y)}{x}}\)?
A. 5
B. 10
C. 15
D. 20
E. Can not be determined

Solution: tough-and-tricky-exponents-and-roots-questions-125956-40.html#p1029231

11. If \(x>0\), \(x^2=2^{64}\) and \(x^x=2^y\) then what is the value of \(y\)?
A. 2
B. 2^(11)
C. 2^(32)
D. 2^(37)
E. 2^(64)

Solution: tough-and-tricky-exponents-and-roots-questions-125956-40.html#p1029232
_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.

What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics

Intern

Joined: 27 Oct 2014

Posts: 8

Re: NEW!!! Tough and tricky exponents and roots questions [#permalink]

Show Tags

07 Dec 2014, 11:20

This post was
BOOKMARKED

Bunuel wrote:

5. If \(x=23^2*25^4*27^6*29^8\) and is a multiple of \(26^n\), where \(n\) is a non-negative integer, then what is the value of \(n^{26}-26^n\)?
A. -26
B. -25
C. -1
D. 0
E. 1

\(23^2*25^4*27^6*29^8=odd*odd*odd*odd=odd\) so \(x\) is an odd number. The only way it to be a multiple of \(26^n\) (even number in integer power) is when \(n=0\), in this case \(26^n=26^0=1\) and 1 is a factor of every integer. Thus \(n=0\) --> \(n^{26}-26^n=0^{26}-26^0=0-1=-1\). Must know for the GMAT: \(a^0=1\), for \(a\neq{0}\) - any nonzero number to the power of 0 is 1. Important note: the case of 0^0 is not tested on the GMAT.

Answer: C.

If 13 is the biggest prime factor of 26 and there isn't a 13 prime factor in X, then 26^0 is the only answer.

Is this approach correct?

Many thanks,

Math Expert

Joined: 02 Sep 2009

Posts: 44599

Re: NEW!!! Tough and tricky exponents and roots questions [#permalink]

Show Tags

08 Dec 2014, 03:10

Expert's post

joaogallegomoura wrote:

Bunuel wrote:

If 13 is the biggest prime factor of 26 and there isn't a 13 prime factor in X, then 26^0 is the only answer.

Is this approach correct?

Many thanks,

______________
Yes, that's correct.
_________________

Manager

Joined: 16 Jan 2013

Posts: 99

Location: Bangladesh

GMAT 1: 490 Q41 V18

GMAT 2: 610 Q45 V28

GPA: 2.75

Re: NEW!!! Tough and tricky exponents and roots questions [#permalink]

Show Tags

10 Jul 2015, 17:55

Note that we need approximate value of the given expression. Now, \(22^{22x}\) is much larger number than \(22^{2x}\). Hence \(22^{22x}-22^{2x}\) will be very close to \(22^{22x}\) itself, basically \(22^{2x}\) is negligible in this case. The same way \(11^{11x}-11^x\) will be very close to \(11^{11x}\) itself.

Hello Bunuel,
Nice explanations. I am a little confused with the fact which you mentioned as "negligible" in this math. Wouldn't this deduction change the result of the answer?

Thanks.

_________________

Heading towards perfection>>

Math Expert

Joined: 02 Sep 2009

Posts: 44599

Re: NEW!!! Tough and tricky exponents and roots questions [#permalink]

Show Tags

11 Jul 2015, 02:40

This post received
KUDOS

Expert's post

This post was
BOOKMARKED

ranaazad wrote:

The question asks about the closest value of the fraction among the options, not the exact value, so we can approximate.

Similar questions to practice:
the-value-of-10-8-10-2-10-7-10-3-is-closest-to-which-of-95082.html
which-of-the-following-is-closest-to-10180-1030-a-110224.html
which-of-the-following-best-approximates-the-value-of-q-if-99674.html
new-tough-and-tricky-exponents-and-roots-questions-125956-40.html
if-x-10-10-x-2-2x-7-3x-2-10x-2-is-closest-to-143897.html
1-0-0001-0-04-10-the-value-of-the-expression-above-is-59398.html
which-of-the-following-is-closest-in-value-to-64425.html
10-180-10-30-which-of-the-following-best-approximates-84309.html
if-x-2-b-8-8-8-6-for-which-of-the-following-b-values-160197.html
if-x-3-000-then-the-value-of-x-2x-1-is-closest-to-166128.html

Hope it helps.
_________________

Director

Joined: 04 Jun 2016

Posts: 617

GMAT 1: 750 Q49 V43

NEW!!! Tough and tricky exponents and roots questions [#permalink]

Show Tags

29 Jun 2016, 11:07

OMG .. there is a much elegant and easier method to solve this

\(\sqrt{6} = ~2.4 ==> 10\sqrt{6}=10*2.4==> 24\)

25+24=49

\(\sqrt{49} =7\)

25-24=1

\(\sqrt{1} = 1\)

So our question is 7+1 = 8

See option \(2 \sqrt{15} = 2*4= 8\)

HENCE C

Bunuel wrote:

SOLUTIONS:

1. What is the value of \(\sqrt{25+10\sqrt{6}}+\sqrt{25-10\sqrt{6}}\)?
A. \(2\sqrt{5}\)
B. \(\sqrt{55}\)
C. \(2\sqrt{15}\)
D. 50
E. 60

Square the given expression to get rid of the roots, though don't forget to un-square the value you get at the end to balance this operation and obtain the right answer:

Must know fro the GMAT: \((x+y)^2=x^2+2xy+y^2\) (while \((x-y)^2=x^2-2xy+y^2\)).

So we get: \((\sqrt{25+10\sqrt{6}}+\sqrt{25-10\sqrt{6}})^2=(\sqrt{25+10\sqrt{6}})^2+2(\sqrt{25+10\sqrt{6}})(\sqrt{25-10\sqrt{6}})+(\sqrt{25-10\sqrt{6}})^2=\)
\(=(25+10\sqrt{6})+2(\sqrt{25+10\sqrt{6}})(\sqrt{25-10\sqrt{6}})+(25-10\sqrt{6})\).

Note that sum of the first and the third terms simplifies to \((25+10\sqrt{6})+(25-10\sqrt{6})=50\), so we have \(50+2(\sqrt{25+10\sqrt{6}})(\sqrt{25-10\sqrt{6}})\) --> \(50+2(\sqrt{25+10\sqrt{6}})(\sqrt{25-10\sqrt{6}})=50+2\sqrt{(25+10\sqrt{6})(25-10\sqrt{6})}\).

Also must know for the GMAT: \((x+y)(x-y)=x^2-y^2\), thus \(50+2\sqrt{(25+10\sqrt{6})(25-10\sqrt{6})}=50+2\sqrt{25^2-(10\sqrt{6})^2)}=50+2\sqrt{625-600}=50+2\sqrt{25}=60\).

Recall that we should un-square this value to get the right the answer: \(\sqrt{60}=2\sqrt{15}\).

Answer: C.

OMG .. there is a much elegant and easier method to solve this

\(\sqrt{6} = ~2.4 ==> 10\sqrt{6}=10*2.4==> 24\)
25+24=49
\(\sqrt{49} =7\)
25-24=1
\(\sqrt{1} = 1\)

So our question is 7+1 = 8

See option \(2 \sqrt{15} = 2*4= 8\)

HENCE C
_________________

Posting an answer without an explanation is "GOD COMPLEX". The world doesn't need any more gods. Please explain you answers properly.
FINAL GOODBYE :- 17th SEPTEMBER 2016. .. 16 March 2017 - I am back but for all purposes please consider me semi-retired.

Intern

Joined: 02 Dec 2017

Posts: 10

Re: NEW!!! Tough and tricky exponents and roots questions [#permalink]

Show Tags

14 Feb 2018, 09:40

Bunuel wrote:

3. If \(5^{10x}=4,900\) and \(2^{\sqrt{y}}=25\) what is the value of \(\frac{(5^{(x-1)})^5}{4^{-\sqrt{y}}}\)?
A. 14/5
B. 5
C. 28/5
D. 13
E. 14

First thing one should notice here is that \(x\) and \(y\) must be some irrational numbers (4,900 has other primes then 5 in its prime factorization and 25 doesn't have 2 as a prime at all), so we should manipulate with given expressions rather than to solve for x and y.

\(5^{10x}=4,900\) --> \((5^{5x})^2=70^2\) --> \(5^{5x}=70\)

\(\frac{(5^{(x-1)})^5}{4^{-\sqrt{y}}}=5^{(5x-5)}*4^{\sqrt{y}}=5^{5x}*5^{-5}*(2^{\sqrt{y}})^2=70*5^{-5}*25^2=70*5^{-5}*5^4=70*5^{-1}=\frac{70}{5}=14\)

Answer: E.

Hey there, just one remark. Why could 4sqrt/y not become by 2 x 2 sqrt/y? why is it definitely (2 sqrt/y)^2 ?

thanks

Math Expert

Joined: 02 Sep 2009

Posts: 44599

Re: NEW!!! Tough and tricky exponents and roots questions [#permalink]

Show Tags

14 Feb 2018, 09:47

This post received
KUDOS

Expert's post

gmatmo wrote:

Bunuel wrote:

Hey there, just one remark. Why could 4sqrt/y not become by 2 x 2 sqrt/y? why is it definitely (2 sqrt/y)^2 ?

thanks

You can write this in several ways:

\(4^{\sqrt{y}}=(2*2)^{\sqrt{y}}=2^{\sqrt{y}}*2^{\sqrt{y}}=2^{\sqrt{y}+\sqrt{y}}=2^{2\sqrt{y}}=(2^{\sqrt{y}})^2\)

P.S. Please read the following post. Writing Mathematical Formulas on the Forum: https://gmatclub.com/forum/rules-for-po ... l#p1096628
_________________

Intern

Joined: 27 May 2017

Posts: 2

NEW!!! Tough and tricky exponents and roots questions [#permalink]

Show Tags

08 Apr 2018, 10:57

Bunuel wrote:

2. What is the units digit of \((17^3)^4-1973^{3^2}\)?
A. 0
B. 2
C. 4
D. 6
E. 8

Must know for the GMAT:
I. The units digit of \((abc)^n\) is the same as that of \(c^n\), which means that the units digit of \((17^3)^4\) is that same as that of \((7^3)^4\) and the units digit of \(1973^{3^2}\) is that same as that of \(3^{3^2}\).

II. If exponentiation is indicated by stacked symbols, the rule is to work from the top down, thus:
\(a^m^n=a^{(m^n)}\) and not \((a^m)^n\), which on the other hand equals to \(a^{mn}\).

So:
\((a^m)^n=a^{mn}\);

\(a^m^n=a^{(m^n)}\).

Thus, \((7^3)^4=7^{(3*4)}=7^{12}\) and \(3^{3^2}=3^{(3^2)}=3^9\).

III. The units digit of integers in positive integer power repeats in specific pattern (cyclicity): The units digit of 7 and 3 in positive integer power repeats in patterns of 4:

1. 7^1=7 (last digit is 7)
2. 7^2=9 (last digit is 9)
3. 7^3=3 (last digit is 3)
4. 7^4=1 (last digit is 1)
5. 7^5=7 (last digit is 7 again!)
...

1. 3^1=3 (last digit is 3)
2. 3^2=9 (last digit is 9)
3. 3^3=27 (last digit is 7)
4. 3^4=81 (last digit is 1)
5. 3^5=243 (last digit is 3 again!)
...

Thus th units digit of \(7^{12}\) will be 1 (4th in pattern, as 12 is a multiple of cyclicty number 4) and the units digit of \(3^9\) will be 3 (first in pattern, as 9=4*2+1).

So, we have that the units digit of \((17^3)^4=17^{12}\) is 1 and the units digit of \(1973^3^2=1973^9\) is 3. Also notice that the second number is much larger then the first one, thus their difference will be negative, something like 11-13=-2, which gives the final answer that the units digit of \((17^3)^4-1973^{3^2}\) is 2.
Answer B.

========================================================================================================================

Hi Bunuel

I have a doubt here.
If we are doing this subtraction (1-3). And, if we would have actually solved the whole subtraction of the number. the we would have borrowed 10 from the tens digit. So, the final subtraction would have looked like (11-3)= 8
please let me know if I am correct or not. if not, then why?

Math Expert

Joined: 02 Sep 2009

Posts: 44599

Re: NEW!!! Tough and tricky exponents and roots questions [#permalink]

Show Tags

08 Apr 2018, 11:35

This post received
KUDOS

Expert's post

shivangibh wrote:

Bunuel wrote:

You could very easily test cases to prove your logic.

(positive number ending with 1) - (greater number ending with 3) = (negative number ending with 2)

11-23=-12
31-133=-102
41-123=-82
....

I tried to explain this several times in this thread:
https://gmatclub.com/forum/new-tough-an ... l#p1054715
https://gmatclub.com/forum/new-tough-an ... l#p1099223

In addition here is an actual result:

\((17^3)^4-1973^{3^2}=-453047530293560259230589943252\)
_________________

Intern

Joined: 28 Nov 2017

Posts: 33

Location: Uzbekistan

NEW!!! Tough and tricky exponents and roots questions [#permalink]

Show Tags

12 Apr 2018, 23:51

1. What is the value of \(\sqrt{25+10\sqrt{6}}+\sqrt{25-10\sqrt{6}}\)?
A. \(2\sqrt{5}\)
B. \(\sqrt{55}\)
C. \(2\sqrt{15}\)
D. 50
E. 60

ALTERNATIVE SOLUTION.
Let \(\sqrt{25+10\sqrt{6}}+\sqrt{25-10\sqrt{6}}\)=x.
Then \(x^2\)=(\(\sqrt{25+10\sqrt{6}}+\sqrt{25-10\sqrt{6}})^2\)
And \(x^2\)=\(25+10\sqrt{6}+2\sqrt{625-600}+25-\sqrt{6}\)
\(x^2=60\) => x=\(2\sqrt{15}\)
Hence C.
_________________

Kindest Regards!
Tulkin.

NEW!!! Tough and tricky exponents and roots questions [#permalink] 12 Apr 2018, 23:51

Print view

First unread post

Go to page Previous 1 2 3 4 5 [ 90 posts ]

Main navigation

GMAT Resources

Partners

MBA Resources

GMAT ® is a registered trademark of the Graduate Management Admission Council ® (GMAC ®). GMAT Club's website has not been reviewed or endorsed by GMAC.

www.gmac.com|www.mba.com |

| GMAT Club Rules | Terms and Conditions

Messages and Replies

Applicant Notifications

Global notifications

GMAT Courses

Private Tutoring

Admissions Consulting

Top 20 by # of applicants

Top 40

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.

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

Request Expert Reply

Events & Promotions

Events & Promotions in June

NEW!!! Tough and tricky exponents and roots questions

Hide Tags

Show Tags

Show Tags

Show Tags

Show Tags

Show Tags

Show Tags

Show Tags

Show Tags

Show Tags

Show Tags

Show Tags

NEW!!! Tough and tricky exponents and roots questions

10 Latest Posts

Main navigation

GMAT Resources

Partners

MBA Resources

Messages and Replies

Applicant Notifications

Global notifications

GMAT Courses

Private Tutoring

Admissions Consulting

Top 20 by # of applicants

Top 40

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.

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

Hello Guest!

Signing up is free, quick, and confidential. Join 700,000+ members and get the full benefits of GMAT Club

Registration gives you:

Tests

Applicant Stats

Books/Downloads

and many more benefits!

Find us on Facebook

Request Expert Reply

Events & Promotions

Events & Promotions in June

NEW!!! Tough and tricky exponents and roots questions

Hide Tags

Show Tags

Show Tags

Show Tags

Show Tags

Show Tags

Show Tags

Show Tags

Show Tags

Show Tags

Show Tags

Show Tags

NEW!!! Tough and tricky exponents and roots questions

10 Latest Posts

Main navigation

GMAT Resources

Partners

MBA Resources

Signing up is free, quick, and confidential.
Join 700,000+ members and get the full benefits of GMAT Club