Question

if x is equal to 3 minus 2 root 2 then find the value of x 4 minus one upon x 4

Answer 1

Given , x = 3 - 2√2 , Find x^4 - 1 / x^4

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solve 'x' for x^4 :
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x = 3 - 2√2

on squaring both side , we get

( x )^2 = ( 3 - 2√2 )^2

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use identity :

( a - b )^2 = a^2 + b^2 - 2 ab
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x^2 = ( 3 )^2 + ( 2√2 )^2 - 2 ( 3 ) ( 2√2 )

x^2 = 9 + 8 - 12√2

x^2 = 17 - 12 √2

again , on squaring both side , we get

( x^2 )^2 = ( 17 - 12√2 )^2

x^4 = 289 + 288 - 408 √2

x^4 = 577 - 408√2

so now

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solve for x^4 - 1/x^4 :
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x^4 - 1 / x^4
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= ( 577 - 408√2 ) - 1 / (577 - 408√2)

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= [ ( 577 - 408√2 )^2 - 1 ] / (577- 408√2 )

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= ( 332929 + 332928 - 470832√2 - 1) / (577 - 408√2 )

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= 665856 - 470832√2 / ( 577 - 408√2 )

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