Author 
Message 
TAGS:

Hide Tags

Manager
Joined: 27 Feb 2010
Posts: 70
Location: Denver

If M = 4^(1/2) + 4^(1/3) + 4^(1/4), then the value of M is:
[#permalink]
Show Tags
26 Apr 2010, 21:07
Question Stats:
65% (01:15) correct 35% (01:20) wrong based on 1743 sessions
HideShow timer Statistics
If \(M=\sqrt{4}+\sqrt[3]{4}+\sqrt[4]{4}\), then the value of M is: A. Less than 3 B. Equal to 3 C. Between 3 and 4 D. Equal to 4 E. Greater than 4
Official Answer and Stats are available only to registered users. Register/ Login.




Math Expert
Joined: 02 Sep 2009
Posts: 64891

If M = 4^(1/2) + 4^(1/3) + 4^(1/4), then the value of M is:
[#permalink]
Show Tags
27 Apr 2010, 07:06
zz0vlb wrote: If \(M=\sqrt{4}+\sqrt[3]{4}+\sqrt[4]{4}\), then the value of M is:
A. less than 3 B. equal to 3 C. between 3 and 4 D. equal to 4 E. greater than 4 Here is a little trick: any positive integer root from a number more than 1 will be more than 1. For instance: \(\sqrt[1000]{2}>1\). Hence \(\sqrt[3]{4}>1\) and \(\sqrt[4]{4}>1\) > \(M=\sqrt{4}+\sqrt[3]{4}+\sqrt[4]{4}=2+(number \ more \ than \ 1)+(number \ more \ than \ 1)>4\) Answer: E.
_________________




Retired Moderator
Joined: 02 Sep 2010
Posts: 702
Location: London

Re: If M = 4^(1/2) + 4^(1/3) + 4^(1/4), then the value of M is:
[#permalink]
Show Tags
23 Sep 2010, 11:09
udaymathapati wrote: Please explain the answer. Attachment: Image2.JPG \(M=4^{1/2} + 4^{1/3} + 4^{1/4}\) Now we know that \(4^{1/2} = 2\) We also know that \(4^{1/4} = \sqrt{2} \approx 1.414 > 1\) And finally \(4^{1/3} > 4^{1/4} \Rightarrow 4^{1/3}>1\) So combining all three together \(M > 2+1+1 \Rightarrow M > 4\)
_________________




Manager
Joined: 01 Feb 2010
Posts: 162

Re: If M = 4^(1/2) + 4^(1/3) + 4^(1/4), then the value of M is:
[#permalink]
Show Tags
26 Apr 2010, 21:52
zz0vlb wrote: Find the value of M (see attachment). I want to see other approaches to this problem.
Source: GMAT Prep sqrt(4) = 2 sqrt(sqrt(4)) = 1.414 approx hence sqrt(4) + sqrt(sqrt(4)) = 3.414 cuberoot(4) > 1 atleast hence answer is M>4.



Intern
Joined: 23 Feb 2012
Posts: 8

Re: If M = 4^(1/2) + 4^(1/3) + 4^(1/4), then the value of M is:
[#permalink]
Show Tags
21 Apr 2012, 12:14
I was wondering why we are not considering the negative roots of 4 in this case. For instance, sq root of 4 would be 2 and 2...same for the 4th root... Any rule / trick that I might be missing here? Look forward to the answer. Cheers!
_________________
Hardwork wont be unrewarded forever!



Math Expert
Joined: 02 Sep 2009
Posts: 64891

Re: If M = 4^(1/2) + 4^(1/3) + 4^(1/4), then the value of M is:
[#permalink]
Show Tags
21 Apr 2012, 12:18
immune wrote: I was wondering why we are not considering the negative roots of 4 in this case. For instance, sq root of 4 would be 2 and 2...same for the 4th root... Any rule / trick that I might be missing here?
Look forward to the answer.
Cheers! Welcome to GMAT Club. Below might help to clear your doubts. 1. GMAT is dealing only with Real Numbers: Integers, Fractions and Irrational Numbers. 2. Any nonnegative real number has a unique nonnegative square root called the principal square root and unless otherwise specified, the square root is generally taken to mean the principal square root. 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, \(\sqrt{25}=+5\) and \(\sqrt{25}=5\). Even roots have only nonnegative value on the GMAT.Odd roots will have the same sign as the base of the root. For example, \(\sqrt[3]{125} =5\) and \(\sqrt[3]{64} =4\). For more check Number Theory chapter of Math Book: mathnumbertheory88376.html
_________________



SVP
Status: The Best Or Nothing
Joined: 27 Dec 2012
Posts: 1706
Location: India
Concentration: General Management, Technology
WE: Information Technology (Computer Software)

Re: If M = 4^(1/2) + 4^(1/3) + 4^(1/4), then the value of M is:
[#permalink]
Show Tags
10 Oct 2013, 21:14
Did in this way:
4 ^1/2 + 4 ^ 1/3 + 4 ^ 1/4
= 4 ^1/2 ( 1 + 4 ^2/3 + 4 ^1/2)
= 2 (1 + 2 + 4 ^2/3)
The above value is obviously above 6, so answer = E



Math Expert
Joined: 02 Sep 2009
Posts: 64891

Re: If M = 4^(1/2) + 4^(1/3) + 4^(1/4), then the value of M is:
[#permalink]
Show Tags
11 Oct 2013, 04:13
somalwar wrote: Did in this way:
4 ^1/2 + 4 ^ 1/3 + 4 ^ 1/4
= 4 ^1/2 ( 1 + 4 ^2/3 + 4 ^1/2)
= 2 (1 + 2 + 4 ^2/3)
The above value is obviously above 6, so answer = E Factoring out is not correct: 4^(1/2)*4^(2/3) does not equal to 4^(1/3). 4^(1/2)*4^(1/2) does not equal to 4^(1/4). \(a^n*a^m=a^{n+m}\) not \(a^{nm}\). Hope it helps.
_________________



Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 9133
GPA: 3.82

Re: If M = 4^(1/2) + 4^(1/3) + 4^(1/4), then the value of M is:
[#permalink]
Show Tags
06 Dec 2015, 23:38
Forget conventional ways of solving math questions. In PS, IVY approach is the easiest and quickest way to find the answer. If M=sqrt(4)+cuberoot(4) +sqrt(sqrt(4)) , then the value of M is: A. Less than 3 B. Equal to 3 C. Between 3 and 4 D. Equal to 4 E. Greater than 4 We know that sqrt(4)=2. Since 1^3=1 < (cuberoot(4))^3=4 < 2^3=8, 1<cuberoot(4)<2. Similarly 1^4=4 < (sqrt(sqrt(4)))^4=4 <2^4=16 implies that 1<sqrt(sqrt(4))<2. So 2+1+1<M<2+2+2. M is between 4 and 6. The answer is, therefore, E.
_________________
MathRevolution: Finish GMAT Quant Section with 10 minutes to spareThe oneandonly World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only $79 for 1 month Online Course""Free Resources30 day online access & Diagnostic Test""Unlimited Access to over 120 free video lessons  try it yourself"



GMAT Club Legend
Joined: 11 Sep 2015
Posts: 4947
Location: Canada

Re: If M = 4^(1/2) + 4^(1/3) + 4^(1/4), then the value of M is:
[#permalink]
Show Tags
07 Dec 2015, 09:23
Quote: If M = √4 + ∛4 + ∜4, then the value of M is:
A) less than 3 B) equal to 3 C) between 3 and 4 D) equal to 4 E) greater than 4
√4√4 = 2 ∛4∛1 = 1 ∛8 = 2 So, ∛4 is BETWEEN 1 and 2. In other words, ∛4 = 1.something∜4∜1 = 1 ∜16 = 2 So, ∜ is BETWEEN 1 and 2. In other words, ∜ = 1.somethingSo, √4 + ∛4 + ∜4 = 2 + 1.something + 1.something= more than 4 = E Cheers, Brent
_________________
Test confidently with gmatprepnow.com



VP
Joined: 09 Mar 2016
Posts: 1252

Re: If M = 4^(1/2) + 4^(1/3) + 4^(1/4), then the value of M is:
[#permalink]
Show Tags
17 Feb 2018, 13:30
shrouded1 wrote: udaymathapati wrote: Please explain the answer. Attachment: Image2.JPG \(M=4^{1/2} + 4^{1/3} + 4^{1/4}\) Now we know that \(4^{1/2} = 2\) We also know that \(4^{1/4} = \sqrt{2} \approx 1.414 > 1\) And finally \(4^{1/3} > 4^{1/4} \Rightarrow 4^{1/3}>1\) So combining all three together \(M > 2+1+1 \Rightarrow M > 4\) How can it \(4^{1/4}\) be \(\sqrt{2}\) what function does exponent 1/4 have \(\sqrt{2}\) without 1/4 exponent equals aprox 1.414



Math Expert
Joined: 02 Sep 2009
Posts: 64891

Re: If M = 4^(1/2) + 4^(1/3) + 4^(1/4), then the value of M is:
[#permalink]
Show Tags
17 Feb 2018, 14:12
dave13 wrote: shrouded1 wrote: udaymathapati wrote: Please explain the answer. Attachment: Image2.JPG \(M=4^{1/2} + 4^{1/3} + 4^{1/4}\) Now we know that \(4^{1/2} = 2\) We also know that \(4^{1/4} = \sqrt{2} \approx 1.414 > 1\) And finally \(4^{1/3} > 4^{1/4} \Rightarrow 4^{1/3}>1\) So combining all three together \(M > 2+1+1 \Rightarrow M > 4\) How can it \(4^{1/4}\) be \(\sqrt{2}\) what function does exponent 1/4 have \(\sqrt{2}\) without 1/4 exponent equals aprox 1.414 \(4^{\frac{1}{4}}=(2^2)^{\frac{1}{4}}=2^{\frac{2}{4}}=2^{\frac{1}{2}}=\sqrt{2}\)
_________________



Director
Joined: 17 Dec 2012
Posts: 628
Location: India

Re: If M = 4^(1/2) + 4^(1/3) + 4^(1/4), then the value of M is:
[#permalink]
Show Tags
08 Mar 2018, 15:19
zz0vlb wrote: If \(M=\sqrt{4}+\sqrt[3]{4}+\sqrt[4]{4}\), then the value of M is:
A. Less than 3 B. Equal to 3 C. Between 3 and 4 D. Equal to 4 E. Greater than 4 Main idea:Approximate RHS by taking the cube root. Details: We have M= 4^ (1/3) + 4^ (1/3)+ 4^ (1/3) which is greater than 4 Hence E.
_________________
Srinivasan Vaidyaraman Magical LogiciansHolistic and Holy Approach



Target Test Prep Representative
Status: Head GMAT Instructor
Affiliations: Target Test Prep
Joined: 04 Mar 2011
Posts: 2799

Re: If M = 4^(1/2) + 4^(1/3) + 4^(1/4), then the value of M is:
[#permalink]
Show Tags
12 Mar 2018, 15:35
zz0vlb wrote: If \(M=\sqrt{4}+\sqrt[3]{4}+\sqrt[4]{4}\), then the value of M is:
A. Less than 3 B. Equal to 3 C. Between 3 and 4 D. Equal to 4 E. Greater than 4 We are given that M = √4 + ^3√4 + ^4√4. We need to determine the approximate value of M. Since √4 = 2, we need to determine the value of 2 + ^3√4 + ^4√4 Let’s determine the approximate value of ^3√4. To find this value, we need to find the perfect cube roots just below and just above the cube root of 4. ^3√1 < ^3√4 < ^3√8 1 < ^3√4 < 2 Let’s next determine the approximate value of ^4√4. To find this value, we need to find the perfect fourth roots just below and just above the fourth root of 4. ^4√1 < ^4√4 < ^4√16 1 < ^4√4 < 2 Since both ^3√4 and ^4√4 are greater than 1, so √4 + ^3√4 + ^4√4 > 2 + 1 + 1, and thus, √4 + ^3√4 + ^4√4 > 4. Answer: E
_________________
5star rated online GMAT quant self study course
See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews



Intern
Joined: 06 Mar 2019
Posts: 46

Re: If M = 4^(1/2) + 4^(1/3) + 4^(1/4), then the value of M is:
[#permalink]
Show Tags
11 Jun 2019, 01:35
Bunuel wrote: immune wrote: I was wondering why we are not considering the negative roots of 4 in this case. For instance, sq root of 4 would be 2 and 2...same for the 4th root... Any rule / trick that I might be missing here?
Look forward to the answer.
Cheers! Welcome to GMAT Club. Below might help to clear your doubts. 1. GMAT is dealing only with Real Numbers: Integers, Fractions and Irrational Numbers. 2. Any nonnegative real number has a unique nonnegative square root called the principal square root and unless otherwise specified, the square root is generally taken to mean the principal square root. 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, \(\sqrt{25}=+5\) and \(\sqrt{25}=5\). Even roots have only nonnegative value on the GMAT.Odd roots will have the same sign as the base of the root. For example, \(\sqrt[3]{125} =5\) and \(\sqrt[3]{64} =4\). For more check Number Theory chapter of Math Book: http://gmatclub.com/forum/mathnumbertheory88376.htmlHi Bunuel , In some data sufficiency questions, the answer becomes insufficient because the square root produces a plus and a minus value. I've been roasted many times for making this careless mistake. But in this question, I got it wrong because I considered 2 as a secondary answer in \(\sqrt{4}\) Can you please help me understand why that is? Thank you. Thanks.



Math Expert
Joined: 02 Sep 2009
Posts: 64891

Re: If M = 4^(1/2) + 4^(1/3) + 4^(1/4), then the value of M is:
[#permalink]
Show Tags
11 Jun 2019, 01:39
Diwabag wrote: Bunuel wrote: immune wrote: I was wondering why we are not considering the negative roots of 4 in this case. For instance, sq root of 4 would be 2 and 2...same for the 4th root... Any rule / trick that I might be missing here?
Look forward to the answer.
Cheers! Welcome to GMAT Club. Below might help to clear your doubts. 1. GMAT is dealing only with Real Numbers: Integers, Fractions and Irrational Numbers. 2. Any nonnegative real number has a unique nonnegative square root called the principal square root and unless otherwise specified, the square root is generally taken to mean the principal square root. 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, \(\sqrt{25}=+5\) and \(\sqrt{25}=5\). Even roots have only nonnegative value on the GMAT.Odd roots will have the same sign as the base of the root. For example, \(\sqrt[3]{125} =5\) and \(\sqrt[3]{64} =4\). For more check Number Theory chapter of Math Book: http://gmatclub.com/forum/mathnumbertheory88376.htmlHi Bunuel , In some data sufficiency questions, the answer becomes insufficient because the square root produces a plus and a minus value. I've been roasted many times for making this careless mistake. But in this question, I got it wrong because I considered 2 as a secondary answer in \(\sqrt{4}\) Can you please help me understand why that is? Thank you. Thanks. The square root function CANNOT produce negative result. EVER. \(\sqrt{...}\) is the square root sign, a function (called the principal square root function), which cannot give negative result. So, this sign (\(\sqrt{...}\)) always means nonnegative square root. The graph of the function f(x) = √xNotice that it's defined for nonnegative numbers and is producing nonnegative results. TO SUMMARIZE: When the GMAT provides the square root sign for an even root, such as a square root, fourth root, etc. then the only accepted answer is the nonnegative root. That is: \(\sqrt{9} = 3\), NOT +3 or 3; \(\sqrt[4]{16} = 2\), NOT +2 or 2; Similarly \(\sqrt{\frac{1}{16}} = \frac{1}{4}\), NOT +1/4 or 1/4. Notice that in contrast, the equation \(x^2 = 9\) has TWO solutions, +3 and 3. Because \(x^2 = 9\) means that \(x =\sqrt{9}=3\) or \(x=\sqrt{9}=3\).
_________________



Senior Manager
Joined: 05 Feb 2018
Posts: 440

Re: If M = 4^(1/2) + 4^(1/3) + 4^(1/4), then the value of M is:
[#permalink]
Show Tags
30 Sep 2019, 16:28
zz0vlb wrote: If \(M=\sqrt{4}+\sqrt[3]{4}+\sqrt[4]{4}\), then the value of M is:
A. Less than 3 B. Equal to 3 C. Between 3 and 4 D. Equal to 4 E. Greater than 4 Even if you're not sure about the concept Bunuel describes, you can prove E easily: The third term, \(\sqrt[4]{4} = 4^{1/4} = 2^{2/4} = 2^{1/2}\) or \(\sqrt[]{2} ≈ 1.4\) So the second term has to be greater than 1.4, and thus M has to be greater than 4 (2 + number bigger than 1.4 + 1.4)



Math Expert
Joined: 02 Sep 2009
Posts: 64891

Re: If M = 4^(1/2) + 4^(1/3) + 4^(1/4), then the value of M is:
[#permalink]
Show Tags
09 Oct 2019, 01:26
zz0vlb wrote: If \(M=\sqrt{4}+\sqrt[3]{4}+\sqrt[4]{4}\), then the value of M is:
A. Less than 3 B. Equal to 3 C. Between 3 and 4 D. Equal to 4 E. Greater than 4 Similar but harder question to practice: https://gmatclub.com/forum/newtoughan ... l#p1029227
_________________



Manager
Joined: 22 Nov 2016
Posts: 245
Concentration: Leadership, Strategy

Re: If M = 4^(1/2) + 4^(1/3) + 4^(1/4), then the value of M is:
[#permalink]
Show Tags
04 Nov 2019, 08:42
Can we not just add the powers here? Since the numerator is the same, adding 1/2 + 1/3 + 1/4 is a positive number and 4 raised to any positive number must be greater than 4 itself.




Re: If M = 4^(1/2) + 4^(1/3) + 4^(1/4), then the value of M is:
[#permalink]
04 Nov 2019, 08:42




