If x + 1/x = 4, find the values of

If x + 1/x = 4, find the values of

(i) x² + 1/x²

(ii) x⁴ + 1/x⁴

(iii) x³ + 1/x³

(iv) x – 1/x.

Answer :

(i) We know that

(x + 1/x)² = x² + 1/x² + 2

It can be written as

x² + 1/x² = (x + 1/x)² – 2

Substituting the values

= 4² – 2

= 16 – 2

= 14

(ii) We know that

(x² + 1/x²)² = x⁴ + 1/x⁴ + 2

It can be written as

x⁴ + 1/x⁴ = (x² + 1/x²)² – 2

Substituting the values

= 14² – 2

= 196 – 2

= 194

(iii) We know that

x³ + 1/x³ = (x + 1/x)³ – 3x (1/x) (x + 1/x)

It can be written as

(x + 1/x)³ – 3(x + 1/x) = 4³ – 3 × 4

By further calculation

= 64 – 12

= 52

(iv) We know that

(x – 1/x)² = x² + 1/x² – 2

Substituting the values

= 14 – 2

= 12

So we get

x – 1/x = ± 2√3

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