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

**(i) x ^{2} + 1/x^{2}**

**(ii) x ^{4} + 1/x^{4}**

**(iii) x ^{3} + 1/x^{3}**

**(iv) x – 1/x.**

**Answer :**

##### (i) We know that

(x + 1/x)^{2} = x^{2} + 1/x^{2} + 2

It can be written as

x^{2} + 1/x^{2} = (x + 1/x)^{2} – 2

Substituting the values

= 4^{2} – 2

= 16 – 2

= 14

##### (ii) We know that

**(x ^{2} + 1/x^{2})^{2} = x^{4} + 1/x^{4} + 2**

It can be written as

x^{4} + 1/x^{4} = (x^{2} + 1/x^{2})^{2} – 2

Substituting the values

= 14^{2} – 2

= 196 – 2

= 194

##### (iii) We know that

**x ^{3} + 1/x^{3} = (x + 1/x)^{3} – 3x (1/x) (x + 1/x)**

It can be written as

(x + 1/x)^{3} – 3(x + 1/x) = 4^{3} – 3 × 4

By further calculation

= 64 – 12

= 52

##### (iv) We know that

**(x – 1/x) ^{2} = x^{2} + 1/x^{2} – 2**

Substituting the values

= 14 – 2

= 12

So we get

x – 1/x = ± 2√3

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