If x^1+x^(−1)=5, what is the value of x^4+x^(−4)?

Question

If  x^1 + x^{(−1)} = 5 , what is the value of  x^4 + x^{(−4)} ?

A. 527
B. 546
C. 579
D. 600
E. 625

JeffTargetTestPrep · Accepted Answer

We are given that x^1 + x^(−1) = 5, i.e., x + 1/x = 5. We need to determine the value of x^4 + x^(-4), i.e., x^4 + 1/x^4.

Let’s square both sides of the equation x + 1/x = 5.:

(x + 1/x)^2 = 5^2

x^2 + 2(x)(1/x) + 1/x^2 = 25

x^2 + 2 + 1/x^2 = 25

x^2 + 1/x^2 = 23

Now let’s square the above equation:

(x^2 + 1/x^2)^2 = 23^2

x^4 + 2(x^2)(1/x^2) + 1/x^4 = 529

x^4 + 2 + 1/x^4 = 529

x^4 + 1/x^4 = 527

Answer: A

0akshay0 · Answer

x^1 + x^{(−1)} = 5  ----------------- I

(x+1/x)^2 = x^2 + 1/x^2 + 2

5^2 = x^2 + 1/x^2 + 2  (by using I)

x^2 + 1/x^2 = 23  --------------- II

(x^2+1/x^2)^2 = x^4 + 1/x^4 + 2

23^2 = x^4 + 1/x^4 + 2  (by using II)

x^4 + 1/x^4 = 529 - 2 = 527

Hence option A is correct

streetking · Answer

Official Answer

Direct attempts to solve for x in this problem will run into quadratics that don't factor and horrible non-integers that need to be raised to fourth powers. Instead, let's focus on manipulating the equation to solve for x^4 + x^−4 directly.

Given the similar structure of the given information, it seems reasonable to begin by squaring the equation x^1+x^−1=5. Be careful, though, not to simply square each term; exponents do not distribute over addition. Instead, recognize the special quadratic. We're looking at two terms added and then squared, so this expression fits the form (a+b)^2=a^2+2ab+b^2. Thus our result will be

(x^1+x^−1^2=5^2

(x^1)^2+2(x^1)(x^−1)+(x^−1)^2=25

x^2+2+x^−2=25

x^2+x^−2=23

Now just repeat the process of squaring both sides once more:

(x^2+x^−2)^2=23^2

x^4+2(x^2)(x^−2)+x^−4=23^2

x^4+2+x^−4=23^2

x^4+x^−4=23^2−2

And it's not even really necessary to calculate 23^2 (which turns out to be 529). 23^2 must end in a 9, so 23^2−2 must end in a 7, and the answer has to be A.

rnz · Answer

why do we need to square it? In the response below your original post, it says that "it's reasonable" to do. Can you explain please? Thanks

niks18 · Answer

Hi  rnz

we are given  x^1+x^{-1}  and need to arrive at  x^4+x^{-4} . So squaring the original equation will raise it to power of  2  i.e.  x^2+x^{-2}  and on further squaring this expression we will reach our destination

mekoner · Answer

Can someone please explain how we arrive at (x^2+1/X^2+2)?

(x+1/x)^2 = (x^2+1/X^2+2)?

niks18 · Answer

Hi mekoner there is a very simple formula used here (a+b)^2=a^2+b^2+2ab now instead of a & b use x & \frac{1}{x} here

exc4libur · Answer

x^1 + x^{−1} = 5 
 (x^1 + x^{−1})^2 = 5^2…x^2+x^{-2}+2=25…x^2+x^{-2}=23 
 (x^2+x^{-2})^2=23^2…x^4+x^{-4}+2=529…x^4+x^{-4}=527

Answer (A)

Abhishek009 · Answer

x^1 + x^{(−1)} = 5

Or,  x + \frac{1}{x} = 5

Or,  x^2 + 2 + \frac{1}{x^2} = 25  (Squaring both sides )

Or,  x^2 + \frac{1}{x^2} = 23

Or,  x^4 + \frac{1}{x^4} = 527  (Squaring both sides ) , Hence Answer must be (A)

ThatDudeKnows · Answer

x^1 + x^{(−1)} = x + \frac{1}{x}

x^1 + x^{(−1)} = 5

x + \frac{1}{x} = 5

Square both sides:

(x + \frac{1}{x})(x + \frac{1}{x}) = 25

x^2 + \frac{x}{x}+ \frac{x}{x} + \frac{1}{x^2} = 25

x^2 + 1 + 1 + \frac{1}{x^2} = 25

x^2 + \frac{1}{x^2} = 23

(Bonus if you spot that we squared the right side and then subtracted 2...if we do that again, we get the right answer.)

Square both sides again:

(x^2 + \frac{1}{x^2})(x^2 + \frac{1}{x^2}) = 23^2

x^4 + \frac{x^2}{x^2} + \frac{x^2}{x^2} + \frac{1}{x^4} = 529

x^4 + 1 + 1 + \frac{1}{x^4} = 529

x^4 + \frac{1}{x^4} = 527

Answer choice A.

gmatophobia · Answer

x + (\frac{1}{x}) = 5

Squaring both sides of the equation

x^2 + (\frac{1}{x})^2 + 2 = 5^2

x^2 + (\frac{1}{x})^2 = 23

Squaring both sides of the equation

x^4 + (\frac{1}{x})^4 + 2 = 23^2

x^4 + (\frac{1}{x})^4 = 529 - 2

x^4 + (\frac{1}{x})^4 = 527

Option A

rory1998 · Answer

Hi Jeff,

When do you know that squaring the equation will lead you to the right path? Is there a pattern of question format that allows you to know right away that squaring both sides would be the easiest way to approach the problem? Because for this question, I attacked it with the traditional way by plugging in (5-1/x)=x into the latter equation and that is a lot of work......

Thanks

bumpbot · Answer

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If x^1+x^(−1)=5, what is the value of x^4+x^(−4)?

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