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

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

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27 Mar 2017, 03:54
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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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Re: If x^1+x^(−1)=5, what is the value of x^4+x^(−4)?  [#permalink]

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29 Mar 2017, 10:10
1
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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Re: If x^1+x^(−1)=5, what is the value of x^4+x^(−4)?  [#permalink]

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27 Mar 2017, 04:26
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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13 Sep 2017, 04:05
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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05 Jan 2018, 22:46
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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Re: If x^1+x^(−1)=5, what is the value of x^4+x^(−4)?  [#permalink]

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05 Jan 2018, 22:57
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
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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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06 Jan 2018, 15:54
Can someone please explain how we arrive at (x^2+1/X^2+2)?

(x+1/x)^2 = (x^2+1/X^2+2)?
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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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06 Jan 2018, 20:20
1
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
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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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13 Jul 2019, 23:06
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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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