If x is not equal to zero, and x + 1/x = 3, then what is the value of

Question

If x is not equal to zero, and  x + \frac{1}{x} = 3 , then what is the value of  x^4 + (\frac{1}{x})^4 ?

A. 27
B. 32
C. 47
D. 64
E. 81

puneetkr · Accepted Answer

we can try it with the following method as well
x + 1/x=3
we square both sides so we have x^2 + 1/x^2 +2 = 9
or x^2 + 1/x^2= 7

squaring again we have x^4 + 1/x^4 + 2 = 49
or x^4 + 1/x^4 = 47 
answer =47 (C)

Bunuel · Answer

I'd solve this one with approximation method rather than solving for x (which will give some irrational number) or squaring twice.

x+\frac{1}{x}=3  -->  x\approx{\frac{5}{2}}  -->  \frac{5}{2}+\frac{2}{5}=2.5+0.4=2.9  (so x is a little bit more than 5/2).

x^4+\frac{1}{x^4}=\frac{625}{16}+\frac{16}{625}  -->  \frac{16}{625}  is a negligible value and  \frac{625}{16}\approx{40} . So as x is a little bit more than 5/2 then the answer should be a little bit more than 40 (taking into account the magnitude of 4th power). Best option is 47.

Answer: C.

Splendidgirl666 · Answer

Thanks. But if you approx as x=2.5 and 1/x as 0.5 then x^4+ 1/x^4 gives around 36. So why would I chose 47 as answer rather than 32?

What would be your tip for approx method here?

Bunuel · Answer

First of all it's better to work with a fraction 5/2 then with a decimal 2.5 as it's easier to calculate  \frac{5^4}{2^4}=\frac{625}{16}  then 2.5^4.

Next, 36 is not a good approximation:  x^4=\frac{625}{16}\approx{40}  (as 16*40=640) -->  since  x  is a little bit more than 5/2 then the answer should be a little bit more than 40  (taking into account the magnitude of 4th power). Best option is 47.

Hope it's clear.

scbguy · Answer

I find puneets method much simpler. I don't think it's intuitive but it's much simpler. Secondly, what is the source of this question?

https://gmatclub.com/static/posted_from.png     Posted from    GMAT ToolKit

Bunuel · Answer

As I've mentioned squaring twice is one of the ways to solve and it's actually quite easy and straightforward as puneetkr showed:

Square once:  x^2+2*x*\frac{1}{x}+\frac{1}{x^2}=9  -->  x^2+\frac{1}{x^2}=7 ;

Square again:  x^4+2*x^2*\frac{1}{x^2}+\frac{1}{x^4}=49  -->  x^4+\frac{1}{x^4}=47 .

Answer: C.

The point is that number 5/2 popped-up in my mind in 2 seconds and taking it to the fourth power took another 2 seconds (5^4=625 and 2^4=16), so I arrived to the correct answer in less than 10 seconds. Though if this doesn't work for you, then algebraic way is also perfectly OK and not lengthy at all.

Abhishek009 · Answer

x+\frac{1}{x} = 3

(x+\frac{1}{x})^2   = x^2 + \frac{1}{x^2} + 2 = 9

So,  x^2 + \frac{1}{x^2} = 7

Now,  (x^2 + \frac{1}{x^2})^2 = x^4 + \frac{1}{x^4} + 2 = 7^2

Or,  x^4 + \frac{1}{x^4} + 2 = 49

Or,  x^4 + \frac{1}{x^4} = 47 
  
Hence correct answer will be (C)

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If x is not equal to zero, and x + 1/x = 3, then what is the value of

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If x is not equal to zero, and x + 1/x = 3, then what is the value of

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