If x 0 and x^(3*x^12)=4, what is the value of x ?

Question

If  x > 0  and  x^{(3*x^{12})}=4 , what is the value of  x  ?

A.  \sqrt {2}

B.  \sqrt {2}

C.  \sqrt {2}

D.  \sqrt{3}

E.  \sqrt{2}

Bunuel · Accepted Answer

Official Solution:

If  x > 0  and  x^{(3*x^{12})}=4 , what is the value of  x  ?

A.  \sqrt {2} 
B.  \sqrt {2} 
C.  \sqrt {2} 
D.  \sqrt{3} 
E.  \sqrt{2}

x^{(3*x^{12})}=4 ;
 
Take to the fourth power:
 
 (x^{(3*x^{12})})^4=4^4 ;
 
 x^{(12*x^{12})}=4^4 ;
 
 (x^{12})^{(x^{12})} =4^4 ;
 
 x^{12}=4=2^2 ;
 
 x=(2^2)^{(\frac{1}{12})} ;
 
 x=2^{(\frac{1}{6})} ;
 
 x=\sqrt {2}

Answer: B

gmatophobia · Answer

We can use the option choices to solve this question. The detailed solution is shown in the attached image. If x gt 0.jpg IMO B

Regor60 · Answer

Set X^12 = Y, so X=Y^(1/12)

and so

(Y^(1/12))^3Y = 4, which equals

Y^(Y/12) = 4^(1/3). Raising each side to the 12th power

Y^Y = 4^(12/3)=4^4, so

Y = 4. Since Y = X^12,

4 = X^12, so X = 4^(1/12) =

2^(1/6)

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samc88 · Answer

You made a mistake on the first step on A,B, and C. 12th Root of 2 is 2^(1/12) NOT 2^(1/24). It's not the 12th root of root 2, the notation is only the 12th root of 2.

AYANGOKE · Answer

kindly explain this step 
 (x^{(3*x^{12})})^4=4^4 ; 
 (x^{12})^{(x^{12})} =4^4 ; 
how come the fourth power didn't affect x^{12}?

Bunuel · Answer

Sire.

The point is  (a^b)^c = a^{(bc)} .

Hence,  (x^{(3*x^{12})})^4=x^{(3*x^{12}*4)}=x^{(12*x^{12})} . Next, we can write  x^{(12*x^{12})}  as  (x^{12})^{(x^{12})}  . Observe that, if we apply  (a^b)^c = a^{(bc)}  to  (x^{12})^{(x^{12})}   we get  x^{(12*x^{12})} .

Hope it helps.

P.S. You might find the below links interesting:

8. Exponents and Roots of Numbers 
     Theory   
 Exponents and Roots on the GMAT: Tips and hints 
 Cyclicity and the remainders on the GMAT 
     Questions   
 Units digits, exponents, remainders problems 
 Tough and tricky DS exponents and roots questions with detailed solutions 
 Tough and tricky PS exponents and roots questions with detailed solutions 
 Exponents DS Questions 
 Exponents PS Questions 
 Roots DS Questions 
 Roots PS Questions

Check below for more:
 ALL YOU NEED FOR QUANT ! ! ! 
 Ultimate GMAT Quantitative Megathread

chloe2m · Answer

Hi Bunuel ! Could you please explain how you go from this line to the other: (x^{12})^{(x^{12})} =4^4 ; x^{12}=4=2^2 ; And from this line to this one: x^{12}=4=2^2 ; x=(2^2)^{(\frac{1}{12})} ; Thank you for your help

Bunuel · Answer

In the equation  (x^{12})^{(x^{12})} =4^4 , observe that on the left side, the expression  x^{12}  is repeated, just as the number 4 is repeated on the right side. Therefore, we can infer  x^{12}=4 .

In the next part, where  x^{12}=2^2 , to isolate x, we take the 12th root of both sides, which gives  x=(2^2)^{(\frac{1}{12})} .

Hope it's clear.

soniasw16 · Answer

How do you know you should take to the 4th power, and not any other power?

Bunuel · Answer

We raise both sides to the 4th power to get the same base and exponent form on both sides.

HeytorBB · Answer

Good question, I just don't agree with the 605... it should be 700, easy!

Bunuel · Answer

The difficulty level of a question on the site, after sufficient attempts, is determined  automatically  based on various parameters collected from users' attempts via timer, such as the percentage of correct answers and the time taken to answer the question. So, this is an 605-655 (Medium) Level level question based on our statistics.

If x > 0 and x^(3*x^12)=4, what is the value of x ?

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