January 19, 2019  January 19, 2019 07:00 AM PST 09:00 AM PST Aiming to score 760+? Attend this FREE session to learn how to Define your GMAT Strategy, Create your Study Plan and Master the Core Skills to excel on the GMAT. January 20, 2019 January 20, 2019 07:00 AM PST 07:00 AM PST Get personalized insights on how to achieve your Target Quant Score.
|
Author |
Message |
|
TAGS:
|
|
|
Intern
Joined: 13 Jun 2010
Posts: 16
|
Re: NEW!!! Tough and tricky exponents and roots questions
[#permalink]
Show Tags
 13 Jan 2013, 23:38
Thanks Bunuel for these interesting problems.
With respect to problem number 2
"So, we have that the units digit of is 1 and the units digit of 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 is 2."
I don't understand the above statement. If the last digit is 1 for the smaller number and the last digit is 3 for the larger number.
And we are trying to find different between smaller number - larger number
eg 757571 - 58299374483. We borrow one from the previous digit and make 1 as 11 and then subtract 3 from 11. So shouldn't the last digit be 8.
I am unable to think any other way. Please let me know where I am going wrong.
Thanks
gmatrant
|
|
|
|
|
|
Math Expert
Joined: 02 Sep 2009
Posts: 52285
|
Re: NEW!!! Tough and tricky exponents and roots questions
[#permalink]
Show Tags
14 Jan 2013, 00:20
gmatrant wrote: Thanks Bunuel for these interesting problems.
With respect to problem number 2
"So, we have that the units digit of is 1 and the units digit of 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 is 2."
I don't understand the above statement. If the last digit is 1 for the smaller number and the last digit is 3 for the larger number.
And we are trying to find different between smaller number - larger number
eg 757571 - 58299374483. We borrow one from the previous digit and make 1 as 11 and then subtract 3 from 11. So shouldn't the last digit be 8.
I am unable to think any other way. Please let me know where I am going wrong.
Thanks
gmatrant First of all 757,571 - 58,299,374,483 = -58,298,616,91 2You could consider easier cases: 11-13=-2, 11-23=-12, 21-83=-62, ...
_________________
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: 03 Sep 2011
Posts: 17
|
Re: NEW!!! Tough and tricky exponents and roots questions
[#permalink]
Show Tags
30 Mar 2013, 13:08
You talk about even and odd in this problem. Wouldn't this not be enough? Since 26 has factors of 2 and 13, even if you had added some even numbers in the product it would still not have sufficed? You would need a 13, 39 etc ? Is my reasoning correct? 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.
|
|
|
|
|
|
Math Expert
Joined: 02 Sep 2009
Posts: 52285
|
Re: NEW!!! Tough and tricky exponents and roots questions
[#permalink]
Show Tags
31 Mar 2013, 07:33
jeremystaub28 wrote: You talk about even and odd in this problem. Wouldn't this not be enough? Since 26 has factors of 2 and 13, even if you had added some even numbers in the product it would still not have sufficed? You would need a 13, 39 etc ? Is my reasoning correct? 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. The fact that the product is odd, is already enough to say that it cannot be a multiple of 26^n unless n=0.
_________________
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
|
|
|
|
|
|
Senior Manager
Joined: 07 Sep 2010
Posts: 259
|
Re: NEW!!! Tough and tricky exponents and roots questions
[#permalink]
Show Tags
16 Apr 2013, 21:53
Bunuel wrote:
• \(\sqrt{x^2}=|x|\), when \(x\leq{0}\), then \(\sqrt{x^2}=-x\) and when \(x\geq{0}\), then \(\sqrt{x^2}=x\)
• When the GMAT provides the square root sign for an even root, such as \(\sqrt{x}\) or \(\sqrt[4]{x}\), then the only accepted answer is the positive root.
That is, \(\sqrt{25}=5\), NOT +5 or -5. In contrast, the equation \(x^2=25\) has TWO solutions, +5 and -5. Even roots have only a positive value on the GMAT.
Hi Bunuel, I am well versed with these exponent rules, however I am becoming confused with the above two rules stated- If \(\sqrt{x^2}\) =\(|x|\) then -> \(\sqrt{5^2}\) = \(|5|\), which means that value can be either 5 or -5. However, as per your example, value should be 5 only. How come this is true? Please clarify. Thanks H
|
|
|
|
|
|
Math Expert
Joined: 02 Sep 2009
Posts: 52285
|
Re: NEW!!! Tough and tricky exponents and roots questions
[#permalink]
Show Tags
16 Apr 2013, 23:09
imhimanshu wrote: Bunuel wrote: 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
Hi Bunuel, Request you to please look at my below solution and kindly guide where I am going wrong.- Given: 5x = 125-3y +z ----(1) \(\sqrt{5x} = 5+\sqrt{z-3y}\) Squaring both sides \(5x= (5+\sqrt{z-3y})^2\) \(5x= 125+(z-3y) +10\sqrt{(z-3y)}\) ---------------(2) If you compare equation 1 and 2; then the term \(\sqrt{(z-3y)}\) has to be 0 If I put this value in the \(\sqrt{\frac{45(z-3y)}{x}}\) Then, the value comes out to be \(\sqrt{0}.\). This is Absurd. I know this is incorrect, but would like to know where I am going wrong. Your help will be appreciated. Thanks H When you square \(5x= (5+\sqrt{z-3y})^2\) you should get 5x= 25+10(z-3y)^1/2+(z-3y) (5^2=25 not 125).
_________________
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
|
|
|
|
|
|
Director
Status: Final Lap Up!!!
Affiliations: NYK Line
Joined: 21 Sep 2012
Posts: 934
Location: India
GMAT 1: 410 Q35 V11 GMAT 2: 530 Q44 V20 GMAT 3: 630 Q45 V31
GPA: 3.84
WE: Engineering (Transportation)
|
Re: NEW!!! Tough and tricky exponents and roots questions
[#permalink]
Show Tags
17 Apr 2013, 15:17
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. Hi Bunuel Can u pls explain the step.....(5^(5x))^2 = 70^2 implies 5^5x = 70 does that mean if a^n = b^n, a=b... Archit
|
|
|
|
|
|
Math Expert
Joined: 02 Sep 2009
Posts: 52285
|
Re: NEW!!! Tough and tricky exponents and roots questions
[#permalink]
Show Tags
17 Apr 2013, 15:20
Archit143 wrote: 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. Hi Bunuel Can u pls explain the step.....(5^(5x))^2 = 70^2 implies 5^5x = 70 does that mean if a^n = b^n, a=b... Archit It's very simple, take the square root from \((5^{5x})^2=70^2\) to get \(5^{5x}=70\).
_________________
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
|
|
|
|
|
|
Manager
Joined: 16 Oct 2012
Posts: 50
|
Re: NEW!!! Tough and tricky exponents and roots questions
[#permalink]
Show Tags
20 Jul 2013, 06:44
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. Bunuel, could you please kindly explain how you come up with the 70 here? Many thanks
|
|
|
|
|
|
Math Expert
Joined: 02 Sep 2009
Posts: 52285
|
Re: NEW!!! Tough and tricky exponents and roots questions
[#permalink]
Show Tags
20 Jul 2013, 06:50
BankerRUS wrote: 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. Bunuel, could you please kindly explain how you come up with the 70 here? Many thanks Can you please tell me which step is unclear: \(5^{10x}=4,900\) --> \((5^{5x})^2=70^2\) --> \(5^{5x}=70\).
_________________
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
|
|
|
|
|
|
Manager
Joined: 16 Oct 2012
Posts: 50
|
Re: NEW!!! Tough and tricky exponents and roots questions
[#permalink]
Show Tags
20 Jul 2013, 07:15
Bunuel wrote: BankerRUS wrote: 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. Bunuel, could you please kindly explain how you come up with the 70 here? Many thanks Can you please tell me which step is unclear: \(5^{10x}=4,900\) --> \((5^{5x})^2=70^2\) --> \(5^{5x}=70\). The first step with the 4,900 is clear. I do not understand the following: 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[/m]and how you get to the 70 there.
|
|
|
|
|
|
Math Expert
Joined: 02 Sep 2009
Posts: 52285
|
Re: NEW!!! Tough and tricky exponents and roots questions
[#permalink]
Show Tags
20 Jul 2013, 08:14
BankerRUS wrote: Bunuel wrote: BankerRUS wrote:
Bunuel, could you please kindly explain how you come up with the 70 here? Many thanks
Can you please tell me which step is unclear: \(5^{10x}=4,900\) --> \((5^{5x})^2=70^2\) --> \(5^{5x}=70\). The first step with the 4,900 is clear. I do not understand the following: 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[/m]and how you get to the 70 there. \(5^{5x}=70\). So, this step is clear. \(\frac{(5^{(x-1)})^5}{4^{-\sqrt{y}}}=5^{(5x-5)}*4^{\sqrt{y}}=5^{5x}*5^{-5}*(2^{\sqrt{y}})^2\). Is this step clear? Next, replace \(5^{5x}\) with 70 and \(2^{\sqrt{y}}\) with 25: \(70*5^{-5}*25^2=70*5^{-5}*5^4=70*5^{-1}=\frac{70}{5}=14\). Does this make sense?
_________________
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
Status: Finance Analyst
Affiliations: CPA Australia
Joined: 10 Jul 2012
Posts: 16
Location: Australia
Concentration: Finance, Healthcare
GMAT 1: 470 Q38 V19 GMAT 2: 600 Q44 V34
GPA: 3.5
WE: Accounting (Health Care)
|
Re: NEW!!! Tough and tricky exponents and roots questions
[#permalink]
Show Tags
22 Jul 2013, 08:28
Edit: [edit] [/edit] Bunuel wrote: 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.
Answer: A. Thanks for the explanation. The part where you take 5+4 out of the full equation is still unclear to me - 5+4(5+5^2+5^3+5^4+5^5). Could you please break it down into multiple steps?
_________________
Our deepest fear is not that we are inadequate. Our deepest fear is that we are powerful beyond measure. It is our light not our darkness that most frightens us.
Your playing small does not serve the world. There's nothing enlightened about shrinking so that other people won't feel insecure around you.
It's not just in some of us; it's in everyone. And as we let our own light shine, we unconsciously give other people permission to do the same.
As we are liberated from our own fear, our presence automatically liberates others.
—Marianne Williamson
|
|
|
|
|
|
Math Expert
Joined: 02 Sep 2009
Posts: 52285
|
Re: NEW!!! Tough and tricky exponents and roots questions
[#permalink]
Show Tags
22 Jul 2013, 08:39
vaishnogmat wrote: Edit: [edit] [/edit] Bunuel wrote: 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.
Answer: A. Thanks for the explanation. The part where you take 5+4 out of the full equation is still unclear to me - 5+4(5+5^2+5^3+5^4+5^5). Could you please break it down into multiple steps? 4 is factored out from 4*5+4*5^2+4*5^3+4*5^4+4*5^5, not 5+4 from the whole expression: \(4*5+4*5^2+4*5^3+4*5^4+4*5^5=4(5+5^2+5^3+5^4+5^5)\). So, \(5+(4*5+4*5^2+4*5^3+4*5^4+4*5^5)=5+4(5+5^2+5^3+5^4+5^5)\). Hope it's clear.
_________________
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: 26 Dec 2011
Posts: 6
|
Re: NEW!!! Tough and tricky exponents and roots questions
[#permalink]
Show Tags
17 Dec 2013, 18:51
Bunuel wrote: 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 \(b\) -->\(b=10=\sqrt{z-3y}\)
Finally, \(\sqrt{\frac{45(z-3y)}{x}}=\sqrt{\frac{45*10^2}{45}}=10\).
Answer: B. This is how I went about solving this problem: Given \(\sqrt{5x} - 5 - \sqrt{z-3y}= 0\): Rearrange and square \((\sqrt{5x})^2 = (\sqrt{z-3y}+5)^2\) \(|5x| = z-3y + 25 + 10\sqrt{z-3y}\) Assuming that x is positive and given \(5x=125-3y+z\), set the two equations equal to each other \(125 - 3y + z = z-3y + 25 + 10\sqrt{z-3y}\) \(100 = 10\sqrt{z-3y}\) \(100 = z-3y\)\(5x = 125-3y+z\) \(5x = 125 + 100\) \(x = 45\)Thus by substitution: \(= \sqrt{\frac{45(z-3y)}{x}}\) \(= \sqrt{\frac{45(100)}{45}}\) \(= \sqrt{100}\) \(= 10\) However, notice that I had to assume that |5x| = 5x Why is this permissible in this problem?
|
|
|
|
|
|
Math Expert
Joined: 02 Sep 2009
Posts: 52285
|
Re: NEW!!! Tough and tricky exponents and roots questions
[#permalink]
Show Tags
18 Dec 2013, 01:04
NateTheGreat11 wrote: Bunuel wrote: 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 \(b\) -->\(b=10=\sqrt{z-3y}\)
Finally, \(\sqrt{\frac{45(z-3y)}{x}}=\sqrt{\frac{45*10^2}{45}}=10\).
Answer: B. This is how I went about solving this problem: Given \(\sqrt{5x} - 5 - \sqrt{z-3y}= 0\): Rearrange and square \((\sqrt{5x})^2 = (\sqrt{z-3y}+5)^2\) \(|5x| = z-3y + 25 + 10\sqrt{z-3y}\) Assuming that x is positive and given \(5x=125-3y+z\), set the two equations equal to each other \(125 - 3y + z = z-3y + 25 + 10\sqrt{z-3y}\) \(100 = 10\sqrt{z-3y}\) \(100 = z-3y\)\(5x = 125-3y+z\) \(5x = 125 + 100\) \(x = 45\)Thus by substitution: \(= \sqrt{\frac{45(z-3y)}{x}}\) \(= \sqrt{\frac{45(100)}{45}}\) \(= \sqrt{100}\) \(= 10\) However, notice that I had to assume that |5x| = 5x Why is this permissible in this problem? Even roots from negative numbers are undefined for the GMAT. We have \(\sqrt{5x}\) in one of the expressions, which means that x is non-negative, thus |5x|=5x. Hope it's clear.
_________________
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
Status: Learning
Joined: 07 Aug 2011
Posts: 36
Location: India
Schools: WBUT - Class of 2011
GMAT Date: 01-06-2014
GPA: 2.6
WE: Research (Education)
|
Re: NEW!!! Tough and tricky exponents and roots questions
[#permalink]
Show Tags
11 Jun 2014, 00:46
5^{5x}*5^{-5}*(2^{\sqrt{y}})^2=70*5^{-5}*25^2 How come you arrive to 25 from y? 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.
_________________
If you like my post give me kudos.
Arindam Sur
Researcher, Academian
|
|
|
|
|
|
Math Expert
Joined: 02 Sep 2009
Posts: 52285
|
Re: NEW!!! Tough and tricky exponents and roots questions
[#permalink]
Show Tags
11 Jun 2014, 00:59
arindamsur wrote: 5^{5x}*5^{-5}*(2^{\sqrt{y}})^2=70*5^{-5}*25^2 How come you arrive to 25 from y? 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. \(4^{\sqrt{y}}=(2^2)^{\sqrt{y}}=2^{2*\sqrt{y}}=(2^{\sqrt{y}})^2=25^2\) --> . P.S. Please go this post: rules-for-posting-please-read-this-before-posting-133935.html#p1096628 (Writing Mathematical Formulas on the Forum). Thank you.
_________________
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
Status: Learning
Joined: 07 Aug 2011
Posts: 36
Location: India
Schools: WBUT - Class of 2011
GMAT Date: 01-06-2014
GPA: 2.6
WE: Research (Education)
|
Re: NEW!!! Tough and tricky exponents and roots questions
[#permalink]
Show Tags
11 Jun 2014, 01:09
\((2^{\sqrt{y}})^2=25^2\) Bunuel explain this part please Bunuel wrote: arindamsur wrote: 5^{5x}*5^{-5}*(2^{\sqrt{y}})^2=70*5^{-5}*25^2 How come you arrive to 25 from y? 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. \(4^{\sqrt{y}}=(2^2)^{\sqrt{y}}=2^{2*\sqrt{y}}=(2^{\sqrt{y}})^2=25^2\) --> . P.S. Please go this post: rules-for-posting-please-read-this-before-posting-133935.html#p1096628 (Writing Mathematical Formulas on the Forum). Thank you. _________________
If you like my post give me kudos.
Arindam Sur
Researcher, Academian
|
|
|
|
|
|
Math Expert
Joined: 02 Sep 2009
Posts: 52285
|
Re: NEW!!! Tough and tricky exponents and roots questions
[#permalink]
Show Tags
11 Jun 2014, 01:11
arindamsur wrote: \((2^{\sqrt{y}})^2=25^2\) Bunuel explain this part please Bunuel wrote: 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. \(4^{\sqrt{y}}=(2^2)^{\sqrt{y}}=2^{2*\sqrt{y}}=(2^{\sqrt{y}})^2=25^2\) --> . P.S. Please go this post: rules-for-posting-please-read-this-before-posting-133935.html#p1096628 (Writing Mathematical Formulas on the Forum). Thank you. We are given in the stem that \(2^{\sqrt{y}}=25\), thus \((2^{\sqrt{y}})^2=25^2\).
_________________
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
|
|
|
|
|
|
|
Re: NEW!!! Tough and tricky exponents and roots questions &nbs
[#permalink]
11 Jun 2014, 01:11
|
|
|
|
|
Go to page
Previous
1 2 3 4 5
Next
[ 93 posts ]
|
|
|
|
|
|
close
