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

Question

If  M=\sqrt{4}+\sqrt {4}+\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

Bunuel · Accepted Answer

Here is a little trick: any positive integer root from a number more than 1 will be more than 1.

For instance:  \sqrt {2}>1 .

Hence:

\sqrt {4}>1  and  \sqrt {4}>1

M=\sqrt{4}+\sqrt {4}+\sqrt {4}=2+(number \ more \ than \ 1)+(number \ more \ than \ 1)>4

Answer: E.

shrouded1 · Answer

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

bangalorian2000 · Answer

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.

immune · Answer

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!

Bunuel · Answer

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 {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 {125} =5 and \sqrt {-64} =-4 . For more check Number Theory chapter of Math Book: math-number-theory-88376.html

PareshGmat · Answer

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

Bunuel · Answer

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.

BrentGMATPrepNow · Answer

√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

dave13 · Answer

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

Bunuel · Answer

4^{\frac{1}{4}}=(2^2)^{\frac{1}{4}}=2^{\frac{2}{4}}=2^{\frac{1}{2}}=\sqrt{2}

SVaidyaraman · Answer

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.

JeffTargetTestPrep · Answer

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

Diwabag · Answer

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}

Can you please help me understand why that is?

Thank you.

Thanks.

Bunuel · Answer

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 .

https://gmatclub.com/forum/download/file.php?id=39502 
 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 {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 .

energetics · Answer

Even if you're not sure about the concept Bunuel describes, you can prove E easily:

The third term,  \sqrt {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)

Bunuel · Answer

Similar but harder question to practice: https://gmatclub.com/forum/new-tough-an ... l#p1029227

BrushMyQuant · Answer

Given that  M=\sqrt{4}+\sqrt {4}+\sqrt {4}  and we need to find the value of M

M=\sqrt{4}+\sqrt {4}+\sqrt {4} 
=> M = 2 +  \sqrt {4} + 2^(2/4)  = 2 +  \sqrt {4} + \sqrt {2}

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

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

So,  Answer will be E 
Hope it helps!

anushree01 · Answer

just memorize this :   Any positive integer root of a number more than 1 will be more than 1.

GmatKnightTutor · Answer

M=\sqrt{4}+\sqrt {4}+\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

Hello, people. In a sense, this is a recognition/estimation question.

Basically, noticing early on that the values in the expression go from Large to Small can be super helpful.

Large ->  \sqrt{4}  = 2

Medium - >  \sqrt {4}  is a bit hard to calculate but is clearly more than the third value " \sqrt {4} "

Small - >  \sqrt {4}  =  \sqrt{2}  = ~1.4

Therefore, the total value has to be more than 2 + ~1.4 + ~1.4

(E) is your answer since the total value has to be greater than 4.

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

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

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