Last visit was: 11 Aug 2024, 23:52 It is currently 11 Aug 2024, 23:52
Toolkit
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

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.

# If M = 4^(1/2) + 4^(1/3) + 4^(1/4), then the value of M is:

SORT BY:
Tags:
Show Tags
Hide Tags
Manager
Joined: 27 Feb 2010
Posts: 54
Own Kudos [?]: 2201 [305]
Given Kudos: 14
Location: Denver
Math Expert
Joined: 02 Sep 2009
Posts: 94856
Own Kudos [?]: 648859 [369]
Given Kudos: 86912
Retired Moderator
Joined: 02 Sep 2010
Posts: 613
Own Kudos [?]: 2976 [22]
Given Kudos: 25
Location: London
Q51  V41
General Discussion
Manager
Joined: 01 Feb 2010
Posts: 89
Own Kudos [?]: 137 [11]
Given Kudos: 2
Re: If M = 4^(1/2) + 4^(1/3) + 4^(1/4), then the value of M is: [#permalink]
8
Kudos
3
Bookmarks
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
Intern
Joined: 23 Feb 2012
Posts: 8
Own Kudos [?]: 1 [1]
Given Kudos: 30
Re: If M = 4^(1/2) + 4^(1/3) + 4^(1/4), then the value of M is: [#permalink]
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?

Cheers!
Math Expert
Joined: 02 Sep 2009
Posts: 94856
Own Kudos [?]: 648859 [12]
Given Kudos: 86912
Re: If M = 4^(1/2) + 4^(1/3) + 4^(1/4), then the value of M is: [#permalink]
5
Kudos
7
Bookmarks
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?

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 non-negative 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 non-negative 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: math-number-theory-88376.html
SVP
Joined: 27 Dec 2012
Status:The Best Or Nothing
Posts: 1558
Own Kudos [?]: 7321 [2]
Given Kudos: 193
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]
1
Kudos
1
Bookmarks
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: 94856
Own Kudos [?]: 648859 [2]
Given Kudos: 86912
Re: If M = 4^(1/2) + 4^(1/3) + 4^(1/4), then the value of M is: [#permalink]
2
Kudos
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.
GMAT Club Legend
Joined: 12 Sep 2015
Posts: 6804
Own Kudos [?]: 30995 [15]
Given Kudos: 799
Re: If M = 4^(1/2) + 4^(1/3) + 4^(1/4), then the value of M is: [#permalink]
10
Kudos
4
Bookmarks
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.something

So, √4 + ∛4 + ∜4 = 2 + 1.something + 1.something
= more than 4
= E

Cheers,
Brent
VP
Joined: 09 Mar 2016
Posts: 1142
Own Kudos [?]: 1034 [0]
Given Kudos: 3851
Re: If M = 4^(1/2) + 4^(1/3) + 4^(1/4), then the value of M is: [#permalink]
shrouded1 wrote:
udaymathapati wrote:

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: 94856
Own Kudos [?]: 648859 [1]
Given Kudos: 86912
Re: If M = 4^(1/2) + 4^(1/3) + 4^(1/4), then the value of M is: [#permalink]
1
Kudos
dave13 wrote:
shrouded1 wrote:
udaymathapati wrote:

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: 586
Own Kudos [?]: 1576 [0]
Given Kudos: 20
Location: India
Re: If M = 4^(1/2) + 4^(1/3) + 4^(1/4), then the value of M is: [#permalink]
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.
Target Test Prep Representative
Joined: 04 Mar 2011
Affiliations: Target Test Prep
Posts: 3036
Own Kudos [?]: 6686 [1]
Given Kudos: 1646
Re: If M = 4^(1/2) + 4^(1/3) + 4^(1/4), then the value of M is: [#permalink]
1
Kudos
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.

Intern
Joined: 06 Mar 2019
Posts: 28
Own Kudos [?]: 14 [0]
Given Kudos: 85
Schools: IMD '21
Re: If M = 4^(1/2) + 4^(1/3) + 4^(1/4), then the value of M is: [#permalink]
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?

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 non-negative 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 non-negative 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: https://gmatclub.com/forum/math-number-theory-88376.html

Hi 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 $$\sqrt{4}$$

Thank you.

Thanks.
Math Expert
Joined: 02 Sep 2009
Posts: 94856
Own Kudos [?]: 648859 [1]
Given Kudos: 86912
Re: If M = 4^(1/2) + 4^(1/3) + 4^(1/4), then the value of M is: [#permalink]
1
Kudos
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?

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 non-negative 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 non-negative 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: https://gmatclub.com/forum/math-number-theory-88376.html

Hi 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 $$\sqrt{4}$$

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 non-negative square root.

The graph of the function f(x) = √x

Notice that it's defined for non-negative numbers and is producing non-negative 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 non-negative 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: 308
Own Kudos [?]: 834 [2]
Given Kudos: 325
Re: If M = 4^(1/2) + 4^(1/3) + 4^(1/4), then the value of M is: [#permalink]
1
Kudos
1
Bookmarks
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: 94856
Own Kudos [?]: 648859 [0]
Given Kudos: 86912
Re: If M = 4^(1/2) + 4^(1/3) + 4^(1/4), then the value of M is: [#permalink]
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/new-tough-an ... l#p1029227
Tutor
Joined: 05 Apr 2011
Status:Tutor - BrushMyQuant
Posts: 1813
Own Kudos [?]: 2158 [0]
Given Kudos: 100
Location: India
Concentration: Finance, Marketing
Schools: XLRI (A)
GMAT 1: 700 Q51 V31
GPA: 3
WE:Information Technology (Computer Software)
Re: If M = 4^(1/2) + 4^(1/3) + 4^(1/4), then the value of M is: [#permalink]
Top Contributor
Given that $$M=\sqrt{4}+\sqrt[3]{4}+\sqrt[4]{4}$$ and we need to find the value of M

$$M=\sqrt{4}+\sqrt[3]{4}+\sqrt[4]{4}$$
=> M = 2 + $$\sqrt[3]{4} + 2^(2/4)$$ = 2 + $$\sqrt[3]{4} + \sqrt[2]{2}$$

Now, $$\sqrt[3]{4}$$ will be between 1 and 2 as $$1^3$$ = 1 and $$2^3$$ = 8
$$\sqrt[2]{2}$$ = 1.414

=> M = 2 + 1.414 + number between 1 and 2
=> M > 2 + 1.414 + 1
=> M > 4.414

Hope it helps!
Non-Human User
Joined: 09 Sep 2013
Posts: 34382
Own Kudos [?]: 861 [0]
Given Kudos: 0
Re: If M = 4^(1/2) + 4^(1/3) + 4^(1/4), then the value of M is: [#permalink]
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.
Re: If M = 4^(1/2) + 4^(1/3) + 4^(1/4), then the value of M is: [#permalink]
Moderator:
Math Expert
94854 posts