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

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Bunuel
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If x > 0 and x^(3*x^12)=4, what is the value of x ? [#permalink] New post   29 Sep 2022, 12:45
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If \(x > 0\) and \(x^{(3*x^{12})}=4\), what is the value of \(x\) ?


A. \(\sqrt[12]{2}\)

B. \(\sqrt[6]{2}\)

C. \(\sqrt[3]{2}\)

D. \(\sqrt{3}\)

E. \(\sqrt{2}\)



 


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B
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gmatophobia
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Re: If x > 0 and x^(3*x^12)=4, what is the value of x ? [#permalink] New post   30 Sep 2022, 00:59
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We can use the option choices to solve this question.

The detailed solution is shown in the attached image.

Attachment:
If x gt 0.jpg
If x gt 0.jpg [ 71.24 KiB | Viewed 2932 times ]


IMO B
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Re: If x > 0 and x^(3*x^12)=4, what is the value of x ? [#permalink] New post   30 Sep 2022, 05:09
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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Re: If x > 0 and x^(3*x^12)=4, what is the value of x ? [#permalink] New post   21 Feb 2023, 14:02
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Bunuel wrote:
If \(x > 0\) and \(x^{(3*x^{12})}=4\), what is the value of \(x\) ?


A. \(\sqrt[12]{2}\)

B. \(\sqrt[6]{2}\)

C. \(\sqrt[3]{2}\)

D. \(\sqrt{3}\)

E. \(\sqrt{2}\)



 


Enjoy this brand new question we just created for the GMAT Club Tests.

To get 1,600 more questions and to learn more visit: user reviews | learn more

 



Official Solution:

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

A. \(\sqrt[12]{2}\)
B. \(\sqrt[6]{2}\)
C. \(\sqrt[3]{2}\)
D. \(\sqrt{3}\)
E. \(\sqrt{2}\)


\(x^{(3*x^{12})}=4\);

Take to the fourth power:

\((x^{(3*x^{12})})^4=4^4\);

\(x^{(12*x^{12})}=4^4\);

\((x^{12})^{(x^{12})} =4^4\);

\(x^{12}=4=2^2\);

\(x=(2^2)^{(\frac{1}{12})}\);

\(x=2^{(\frac{1}{6})}\);

\(x=\sqrt[6]{2}\)


Answer: B
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Re: If x > 0 and x^(3*x^12)=4, what is the value of x ? [#permalink] New post   02 Jun 2023, 03:26
gmatophobia wrote:
We can use the option choices to solve this question.

The detailed solution is shown in the attached image.

IMO C


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.
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Re: If x > 0 and x^(3*x^12)=4, what is the value of x ? [#permalink] New post   23 Jun 2023, 15:50
Bunuel wrote:
Bunuel wrote:
If \(x > 0\) and \(x^{(3*x^{12})}=4\), what is the value of \(x\) ?


A. \(\sqrt[12]{2}\)

B. \(\sqrt[6]{2}\)

C. \(\sqrt[3]{2}\)

D. \(\sqrt{3}\)

E. \(\sqrt{2}\)



 


Enjoy this brand new question we just created for the GMAT Club Tests.

To get 1,600 more questions and to learn more visit: user reviews | learn more

 



Official Solution:

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

A. \(\sqrt[12]{2}\)
B. \(\sqrt[6]{2}\)
C. \(\sqrt[3]{2}\)
D. \(\sqrt{3}\)
E. \(\sqrt{2}\)


\(x^{(3*x^{12})}=4\);

Take to the fourth power:

\((x^{(3*x^{12})})^4=4^4\);

\(x^{(12*x^{12})}=4^4\);

\((x^{12})^{(x^{12})} =4^4\);

\(x^{12}=4=2^2\);

\(x=(2^2)^{(\frac{1}{12})}\);

\(x=2^{(\frac{1}{6})}\);

\(x=\sqrt[6]{2}\)


Answer: B



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}?
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Re: If x > 0 and x^(3*x^12)=4, what is the value of x ? [#permalink] New post   24 Jun 2023, 00:58
Expert Reply
AYANGOKE wrote:
Bunuel wrote:
Bunuel wrote:
If \(x > 0\) and \(x^{(3*x^{12})}=4\), what is the value of \(x\) ?


A. \(\sqrt[12]{2}\)

B. \(\sqrt[6]{2}\)

C. \(\sqrt[3]{2}\)

D. \(\sqrt{3}\)

E. \(\sqrt{2}\)



 


Enjoy this brand new question we just created for the GMAT Club Tests.

To get 1,600 more questions and to learn more visit: user reviews | learn more

 



Official Solution:

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

A. \(\sqrt[12]{2}\)
B. \(\sqrt[6]{2}\)
C. \(\sqrt[3]{2}\)
D. \(\sqrt{3}\)
E. \(\sqrt{2}\)


\(x^{(3*x^{12})}=4\);

Take to the fourth power:

\((x^{(3*x^{12})})^4=4^4\);

\(x^{(12*x^{12})}=4^4\);

\((x^{12})^{(x^{12})} =4^4\);

\(x^{12}=4=2^2\);

\(x=(2^2)^{(\frac{1}{12})}\);

\(x=2^{(\frac{1}{6})}\);

\(x=\sqrt[6]{2}\)


Answer: B



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}?


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




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chloe2m
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If x > 0 and x^(3*x^12)=4, what is the value of x ? [#permalink] New post   12 Jan 2024, 12:47
Bunuel wrote:
Bunuel wrote:
If \(x > 0\) and \(x^{(3*x^{12})}=4\), what is the value of \(x\) ?


A. \(\sqrt[12]{2}\)

B. \(\sqrt[6]{2}\)

C. \(\sqrt[3]{2}\)

D. \(\sqrt{3}\)

E. \(\sqrt{2}\)



 


Enjoy this brand new question we just created for the GMAT Club Tests.

To get 1,600 more questions and to learn more visit: user reviews | learn more

 



Official Solution:

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

A. \(\sqrt[12]{2}\)
B. \(\sqrt[6]{2}\)
C. \(\sqrt[3]{2}\)
D. \(\sqrt{3}\)
E. \(\sqrt{2}\)


\(x^{(3*x^{12})}=4\);

Take to the fourth power:

\((x^{(3*x^{12})})^4=4^4\);

\(x^{(12*x^{12})}=4^4\);

\((x^{12})^{(x^{12})} =4^4\);

\(x^{12}=4=2^2\);

\(x=(2^2)^{(\frac{1}{12})}\);

\(x=2^{(\frac{1}{6})}\);

\(x=\sqrt[6]{2}\)


Answer: B


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 :)
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Bunuel
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Re: If x > 0 and x^(3*x^12)=4, what is the value of x ? [#permalink] New post   12 Jan 2024, 13:08
Expert Reply
chloe2m wrote:
Bunuel wrote:
Bunuel wrote:
If \(x > 0\) and \(x^{(3*x^{12})}=4\), what is the value of \(x\) ?


A. \(\sqrt[12]{2}\)

B. \(\sqrt[6]{2}\)

C. \(\sqrt[3]{2}\)

D. \(\sqrt{3}\)

E. \(\sqrt{2}\)



 


Enjoy this brand new question we just created for the GMAT Club Tests.

To get 1,600 more questions and to learn more visit: user reviews | learn more

 



Official Solution:

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

A. \(\sqrt[12]{2}\)
B. \(\sqrt[6]{2}\)
C. \(\sqrt[3]{2}\)
D. \(\sqrt{3}\)
E. \(\sqrt{2}\)


\(x^{(3*x^{12})}=4\);

Take to the fourth power:

\((x^{(3*x^{12})})^4=4^4\);

\(x^{(12*x^{12})}=4^4\);

\((x^{12})^{(x^{12})} =4^4\);

\(x^{12}=4=2^2\);

\(x=(2^2)^{(\frac{1}{12})}\);

\(x=2^{(\frac{1}{6})}\);

\(x=\sqrt[6]{2}\)


Answer: B


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 :)


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.
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Re: If x > 0 and x^(3*x^12)=4, what is the value of x ? [#permalink] New post   13 Jan 2024, 10:41
Hi Bunuel what do you think the correct way of solving this type of question in gmat, I understand both the way but for me option putting is easier, but dont know when it will bad and time consuming, so asking..
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Re: If x > 0 and x^(3*x^12)=4, what is the value of x ? [#permalink]
13 Jan 2024, 10:41
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