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Math Expert V
Joined: 02 Sep 2009
Posts: 58092
If x^1+x^(−1)=5, what is the value of x^4+x^(−4)?  [#permalink]

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17 00:00

Difficulty:   55% (hard)

Question Stats: 61% (01:52) correct 39% (02:24) wrong based on 201 sessions

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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

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Target Test Prep Representative G
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Re: If x^1+x^(−1)=5, what is the value of x^4+x^(−4)?  [#permalink]

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5
Bunuel wrote:
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

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

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##### General Discussion
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Re: If x^1+x^(−1)=5, what is the value of x^4+x^(−4)?  [#permalink]

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1
3
Bunuel wrote:
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

$$x^1 + x^{(−1)} = 5$$ ----------------- 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
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Re: If x^1+x^(−1)=5, what is the value of x^4+x^(−4)?  [#permalink]

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2

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.
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Re: If x^1+x^(−1)=5, what is the value of x^4+x^(−4)?  [#permalink]

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JeffTargetTestPrep wrote:
Bunuel wrote:
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

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

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
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Joined: 25 Feb 2013
Posts: 1191
Location: India
GPA: 3.82
Re: If x^1+x^(−1)=5, what is the value of x^4+x^(−4)?  [#permalink]

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rnz wrote:
JeffTargetTestPrep wrote:
Bunuel wrote:
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

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

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

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
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Joined: 30 Oct 2017
Posts: 8
Re: If x^1+x^(−1)=5, what is the value of x^4+x^(−4)?  [#permalink]

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

(x+1/x)^2 = (x^2+1/X^2+2)?
Retired Moderator D
Joined: 25 Feb 2013
Posts: 1191
Location: India
GPA: 3.82
Re: If x^1+x^(−1)=5, what is the value of x^4+x^(−4)?  [#permalink]

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

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

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 Non-Human User Joined: 09 Sep 2013
Posts: 12371
Re: If x^1+x^(−1)=5, what is the value of x^4+x^(−4)?  [#permalink]

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_________________ Re: If x^1+x^(−1)=5, what is the value of x^4+x^(−4)?   [#permalink] 13 Jul 2019, 23:06
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