**If x + 1/x = 2, prove that x**^{2} + 1/x^{2} = x^{3} + 1/x^{3} = x^{4} + 1/x^{4}.

^{2}+ 1/x

^{2}= x

^{3}+ 1/x

^{3}= x

^{4}+ 1/x

^{4}.

**Answer :**

We know that

**x ^{2} + 1/x^{2} = (x + 1/x)² – 2**

Substituting the values

x^{2} + 1/x^{2} = 2^{2} – 2

So we get

x^{2} + 1/x^{2} = 4 – 2 = 2 …. (1)

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

Substituting the values

x^{3} + 1/x^{3} = 2^{3} – 3 × 2

So we get

x^{3} + 1/x^{3} = 8 – 6 = 2 …… (2)

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

Substituting the values

x^{4} + 1/x^{4} = 2^{2} – 2

So we get

x^{4} + 1/x^{4} = 4 – 2 = 2 …. (3)

From equation (1), (2) and (3)

x^{2} + 1/x^{2} = x^{3} + 1/x^{3} = x^{4} + 1/x^{4}

Hence, it is proved.

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