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.
It appears that you are browsing the GMAT Club forum unregistered!
Signing up is free, quick, and confidential.
Join other 500,000 members and get the full benefits of GMAT Club
Registration gives you:
Tests
Take 11 tests and quizzes from GMAT Club and leading GMAT prep companies such as Manhattan GMAT,
Knewton, and others. All are free for GMAT Club members.
Applicant Stats
View detailed applicant stats such as GPA, GMAT score, work experience, location, application
status, and more
Books/Downloads
Download thousands of study notes,
question collections, GMAT Club’s
Grammar and Math books.
All are free!
Thank you for using the timer!
We noticed you are actually not timing your practice. Click the START button first next time you use the timer.
There are many benefits to timing your practice, including:
NEW!!! Tough and tricky exponents and roots questions [#permalink]
12 Jan 2012, 02:03
50
This post received KUDOS
Expert's post
131
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
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
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
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
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. \(2^{22x}*11^{22x}\)
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
Re: NEW!!! Tough and tricky exponents and roots questions [#permalink]
13 Jan 2012, 20:46
2
This post received KUDOS
Q4. What is the value of \(5+4*5+4*5^2+4*5^3+4*5^4+4*5^5\)
This one was actually the simplest I thought. Here is how:
\(5+4*5+4*5^2+4*5^3+4*5^4+4*5^5\)
So \(5*(1+4)+4*5^2+4*5^3+4*5^4+4*5^5\) So \(5^2+4*5^2+4*5^3+4*5^4+4*5^5\) So \(5^2*(1+4)+4*5^3+4*5^4+4*5^5\) So \(5^3+4*5^3+4*5^4+4*5^5\) So \(5^3*(1+4)+4*5^4+4*5^5\)
So every expression behind contributes a power of 1 to the one in front of it. We just need to see the last which is \(5^5\) Keep solving and you come to a total of \(5^6\)
Hence Answer = \(5^6\) _________________
"Nowadays, people know the price of everything, and the value of nothing."Oscar Wilde
ah, I got what u meant (at last! heh) so, this q. is similar to the q "2001/6001 is close to which of the following?" the answ- 2/6=1/3 _________________
Happy are those who dream dreams and are ready to pay the price to make them come true
I am still on all gmat forums. msg me if you want to ask me smth
Re: NEW!!! Tough and tricky exponents and roots questions [#permalink]
14 Jan 2012, 10:44
2
This post received KUDOS
SonyGmat wrote:
LalaB wrote:
10. Given that 5x=125-3y+z and \sqrt{5x}-25-\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
5x-125=z-3y
sqrt{5x}-25=\sqrt{z-3y} (sqrt{5x}-25)^2=z-3y
(sqrt{5x}-25)^2=5x-125 \sqrt{5x}=15 x=5*9=45
so the expression \sqrt{\frac{45(z-3y)}{x} = \sqrt{45(15-25)/45}=\sqrt{-10}
E is the answ
bunuel wrote:
10. Given that \(5x=125-3y+z\) and \(\sqrt{5x}-25-\sqrt{z-3y}=0\), 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
hm, strange enough. I seem to overtaxed myself . i referred to the 2nd equation, since we need to find sqroot(z-3y) \sqrt{5x}-25=\sqrt{z-3y} \sqrt{5*45}-25=\sqrt{z-3y} -10=\sqrt{z-3y}
but at the same time 5x=125-3y+z 5*45-125=z-3y 10=z-3y it seems there is a typo in a q stem -\sqrt{5x}-25-\sqrt{z-3y}=0
it must be \sqrt{5x}-5-\sqrt{z-3y}=0 _________________
Happy are those who dream dreams and are ready to pay the price to make them come true
I am still on all gmat forums. msg me if you want to ask me smth
Re: NEW!!! Tough and tricky exponents and roots questions [#permalink]
14 Jan 2012, 11:21
Expert's post
LalaB wrote:
SonyGmat wrote:
LalaB wrote:
10. Given that 5x=125-3y+z and \sqrt{5x}-25-\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
5x-125=z-3y
sqrt{5x}-25=\sqrt{z-3y} (sqrt{5x}-25)^2=z-3y
(sqrt{5x}-25)^2=5x-125 \sqrt{5x}=15 x=5*9=45
so the expression \sqrt{\frac{45(z-3y)}{x} = \sqrt{45(15-25)/45}=\sqrt{-10}
E is the answ
bunuel wrote:
10. Given that \(5x=125-3y+z\) and \(\sqrt{5x}-25-\sqrt{z-3y}=0\), 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
hm, strange enough. I seem to overtaxed myself . i referred to the 2nd equation, since we need to find sqroot(z-3y) \sqrt{5x}-25=\sqrt{z-3y} \sqrt{5*45}-25=\sqrt{z-3y} -10=\sqrt{z-3y}
but at the same time 5x=125-3y+z 5*45-125=z-3y 10=z-3y it seems there is a typo in a q stem -\sqrt{5x}-25-\sqrt{z-3y}=0
it must be \sqrt{5x}-5-\sqrt{z-3y}=0
Yes, there is a typo: must be 5 instead of 25. Thanks. Edited. _________________
Re: NEW!!! Tough and tricky exponents and roots questions [#permalink]
14 Jan 2012, 13:53
12
This post received KUDOS
Expert's post
14
This post was BOOKMARKED
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: \(\sqrt{60}=2\sqrt{15}\).
Re: NEW!!! Tough and tricky exponents and roots questions [#permalink]
14 Jan 2012, 13:58
10
This post received KUDOS
Expert's post
13
This post was BOOKMARKED
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.
Re: NEW!!! Tough and tricky exponents and roots questions [#permalink]
14 Jan 2012, 14:00
12
This post received KUDOS
Expert's post
14
This post was BOOKMARKED
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.
Re: NEW!!! Tough and tricky exponents and roots questions [#permalink]
14 Jan 2012, 14:03
8
This post received KUDOS
Expert's post
9
This post was BOOKMARKED
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
This question can be solved in several ways:
Traditional approach: \(5+4*5+4*5^2+4*5^3+4*5^4+4*5^5=5+4(5+5^2+5^3+5^4+5^5)\) Note that we have the sum of geometric progression in brackets with first term equal to 5 and common ratio also equal to 5. The sum of the first \(n\) terms of geometric progression is given by: \(sum=\frac{b*(r^{n}-1)}{r-1}\), where \(b\) is the first term, \(n\) # of terms and \(r\) is a common ratio \(\neq{1}\).
So in our case: \(5+4(5+5^2+5^3+5^4+5^5)=5+4(\frac{5(5^5-1)}{5-1})=5^6\).
30 sec approach based on answer choices: We have the sum of 6 terms. Now, if all terms were equal to the largest term 4*5^5 we would have: \(sum=6*(4*5^5)=24*5^5\approx{5^2*5^5}\approx{5^7}\), so the actual sum must be less than 5^7, thus the answer must be A: 5^6.
Re: NEW!!! Tough and tricky exponents and roots questions [#permalink]
14 Jan 2012, 14:05
13
This post received KUDOS
Expert's post
15
This post was BOOKMARKED
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.
Re: NEW!!! Tough and tricky exponents and roots questions [#permalink]
14 Jan 2012, 14:06
9
This post received KUDOS
Expert's post
8
This post was BOOKMARKED
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
Must know for the GMAT: Even roots from negative number is undefined on the GMAT (as GMAT is dealing only with Real Numbers): \(\sqrt[{even}]{negative}=undefined\), for example \(\sqrt{-25}=undefined\).
Odd roots have the same sign as the base of the root. For example, \(\sqrt[3]{125} =5\) and \(\sqrt[3]{-64} =-4\).
Back to the original question:
As \(-2^5=-32\) then \(x\) must be a little bit less than -2 --> \(x=\sqrt[5]{-37}\approx{-2.1}<-2\). Thus \(x^3\approx{(-2.1)^3}\approx{-8.something}<-8\), so option D must be true.
As for the other options: A. \(\sqrt{-x}=\sqrt{-(-2.1)}=\sqrt{2.1}<2\), \(\sqrt{-x}>2\) is not true. B. \(x\approx{-2.1}<-2\), thus x>-2 is also not true. C. \(x^2\approx{(-2.1)}^2=4.something>4\), thus x^2<4 is also not true. E. \(x^4\approx{(-2.1)}^4\approx17\), (2^4=16, so anyway -2.1^4 can not be more than 32) thus x^4>32 is also not true.
Re: NEW!!! Tough and tricky exponents and roots questions [#permalink]
14 Jan 2012, 14:08
3
This post received KUDOS
Expert's post
15
This post was BOOKMARKED
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
Here is a little trick: any positive integer root from a number more than 1 will be more than 1. For example: \(\sqrt[1000]{2}>1\).
Now, \(\sqrt{10}>3\) (as 3^2=9) and \(\sqrt[3]{9}>2\) (2^3=8). Thus \(x=(# \ more \ then \ 3)+(# \ more \ then \ 2)+(7 \ numbers \ more \ then \ 1)=\) \(=(# \ more \ then \ 5)+(# \ more \ then \ 7)=\) \(=(# \ more \ then \ 12)\)
Re: NEW!!! Tough and tricky exponents and roots questions [#permalink]
14 Jan 2012, 14:09
8
This post received KUDOS
Expert's post
18
This post was BOOKMARKED
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
Given: \(x>0\) and \(x=\sqrt{6+{\sqrt{6+\sqrt{6+\sqrt{6+...}}}}}\) --> \(x=\sqrt{6+({\sqrt{6+\sqrt{6+\sqrt{6+...})}}}}\), as the expression under the square root extends infinitely then expression in brackets would equal to \(x\) itself and we can safely replace it with \(x\) and rewrite the given expression as \(x=\sqrt{6+x}\). Square both sides: \(x^2=6+x\) --> \((x+2)(x-3)=0\) --> \(x=-2\) or \(x=3\), but since \(x>0\) then: \(x=3\).
Re: NEW!!! Tough and tricky exponents and roots questions [#permalink]
14 Jan 2012, 14:11
5
This post received KUDOS
Expert's post
11
This post was BOOKMARKED
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. \(2^{22x}*11^{22x}\)
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.
You can check this algebraically as well: \(\frac{22^{22x}-22^{2x}}{11^{11x}-11^x}=\frac{22^{2x}(22^{20x}-1)}{11^x(11^{10x}-1)}\). Again, -1, both in denominator and nominator is negligible value and we'll get the same expression as above: \(\frac{22^{2x}(22^{20x}-1)}{11^x(11^{10x}-1)}\approx{\frac{22^{2x}*22^{20x}}{11^x*11^{10x}}}=\frac{22^{22x}}{11^{11x}}=2^{22x}*11^{11x}\)
Re: NEW!!! Tough and tricky exponents and roots questions [#permalink]
14 Jan 2012, 14:13
7
This post received KUDOS
Expert's post
16
This post was BOOKMARKED
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
Rearranging both expressions we'll get: \(5x-(z-3y)=125\) and \(\sqrt{5x}-\sqrt{z-3y}=5\). Denote \(\sqrt{5x}\) as \(a\) and \(\sqrt{z-3y}\) as \(b\).
So we have that \(a^2-b^2=125\) and \(a-b=5\). Now, \(a^2-b^2=(a-b)(a+b)=125\) and as \(a-b=5\) then \((a-b)(a+b)=5*(a+b)=125\) --> \(a+b=25\). Thus we get two equations with two unknowns: \(a+b=25\) and \(a-b=5\) --> solving for \(a\) --> \(a=15=\sqrt{5x}\) --> \(x=45\). Solving for \(b\) -->\(b=10=\sqrt{z-3y}\)
As I’m halfway through my second year now, graduation is now rapidly approaching. I’ve neglected this blog in the last year, mainly because I felt I didn’...
Perhaps known best for its men’s basketball team – winners of five national championships, including last year’s – Duke University is also home to an elite full-time MBA...
Hilary Term has only started and we can feel the heat already. The two weeks have been packed with activities and submissions, giving a peek into what will follow...