Question

if x=0 and x+¹/x=2 then show that x²+1/x²=x³+1/x³=x4+1/x4​

Answer 1

Question

If $x\neq 0$ and $x+\dfrac{1}{x} =2$,

show that

$x^{2}+\dfrac{1}{x^{2}} =x^{3}+\dfrac{1}{x^{3}} =x^{4}+\dfrac{1}{x^{4}}$.

Solution

① The value of $x^{2}+\dfrac{1}{x^{2}}$.

Squaring both sides of the given equation,

$\implies (x+\dfrac{1}{x} )^{2}=2^{2}$

$\implies x^{2}+2+\dfrac{1}{x^{2}} =4$

$\implies x^{2}+\dfrac{1}{x^{2}} =2$

② The value of $x^{3}+\dfrac{1}{x^{3}}$.

Cubing both sides of the given equation,

$\implies (x+\dfrac{1}{x} )^{3}=2^{3}$

$\implies x^{3}+3x+\dfrac{3}{x} +\dfrac{1}{x^{3}} =8$

$\implies x^{3}+3(x+\dfrac{1}{x} )+\dfrac{1}{x^{3}} =8$

$\implies x^{3}+3\cdot2+\dfrac{1}{x^{3}} =8$

$\implies x^{3}+\dfrac{1}{x^{3}} =2$

③ The value of $x^{4}+\dfrac{1}{x^{4}}$.

We can double the exponent of $x^{2}+\dfrac{1}{x^{2}}$ by squaring it again.

Let's square both sides.

$\implies (x^{2}+\dfrac{1}{x^{2}} )^{2}=2^{2}$

$\implies x^{4}+2+\dfrac{1}{x^{4}} =4$

$\implies x^{4}+\dfrac{1}{x^{4}} =2$

④ Conclusion.

So,

$x^{2}+\dfrac{1}{x^{2}} =x^{3}+\dfrac{1}{x^{3}} =x^{4}+\dfrac{1}{x^{4}}$

and each expression is equal to 2.