## If x = 7 + 4√3, find the value of x^4 + (1/x^4).

Question

If x = 7 + 4√3, find the value of x^4 + (1/x^4).

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2 weeks 2021-09-07T12:56:27+00:00 2 Answers 0 views 0

x=7+4√3

1/x=1/7+4√3

=1/7+4√3×7-4√3/7-4√3

=7-4√3/1

=7-4√3

x^4+1/x^4

=(7+4√3)^4+(7-4√3)^4

=((7+4√3)^2)^2 +((7-4√3)^2)^2

=(97+56√3)^2+ (97-56√3)^2

=2((97)^2+(56√3)^2)

=2(9409+9408)

=37,634

x^4+1/x^4 =37,634

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_______________________

Given :

x = 7 + 4√3

To find :

√x + 1 / √x

Solution :

x = 7 + 4√3

⇒ x = 4 + 3 + 4√3

⇒ x = 2² + √3² + 2 × 2 × √3

⇒ x = ( 2 + √3)²

⇒ √x = 2 + √3

Now,

1 / √x = 1 / 2 + √3 × 2 – √3 / 2 – √3

⇒ 1 / √x = 2 – √3 / 2² – √3²

⇒ 1 / √x = 2 – √3 / 4 – 3

⇒ 1 / √x = 2 – √3

Again,

√x + 1 / √x = 2 + √3 + 2 – √3

⇒ √x + 1 / √x = 2 + 2

⇒ √x + 1 / √x = 4

Hence,

√x + 1 / √x = 4

_______________________

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