Question

$\rm \: if \: x+\frac {1}{x}=2 \: then \: show \: that \: {x}^{2} + \frac{1}{ {x}^{2} } = {x}^{3} + \frac{1}{ {x}^{3} } = {x}^{4} + \frac{1}{ {x}^{4} }$

Answer 1

Step-by-step explanation:

refer to the attachment:

Answer 2

Answer:

Hello Dear User__________

Here is Your Answer...!!

____________________

Step by step solution:

$Given \ x+\frac{1}{x}=2\\ \\we \ have \ to \ show \ that \ x^{2}+\frac{1}{x^{2} } =x^{3}+\frac{1}{x^{3}}=x^4+\frac{1}{x^{4}}\\\\Let \ us \ do\\\\First \ squaring \ in \ given \ equation\\\\( x+\frac{1}{x})^2=(2)^2\\\\ x^{2}+\frac{1}{x^{2} }+2=4\\\\ x^{2}+\frac{1}{x^{2} }=2\\\\Now \ cubing \ both \ side \ in \ given \ equation\\\\(x+\frac{1}{x})^3=(2)^3\\\\x^{3}+\frac{1}{x^{3}}+3(x+\frac{1}{x})(x \times \frac{1}{x})=8\\\\putting \ value \ of \ (x+\frac{1}{x})=2 \ here$

$x^{3}+\frac{1}{x^{3}}+3 \times 2 \times1=8\\\\x^{3}+\frac{1}{x^{3}}=8-6\\\\x^{3}+\frac{1}{x^{3}}=2\\\\Now \ squaring \ on \ both \ side \ in \ (x^{2}+\frac{1}{x^{2}})=2\\\\(x^{2}+\frac{1}{x^{2}})^2=(2)^2\\\\ (x^{4}+\frac{1}{x^{4}})+2=4\\\\(x^{4}+\frac{1}{x^{4}})=2\\\\From \ here \ we \ shown\\\\x^{2}+\frac{1}{x^{2} } =x^{3}+\frac{1}{x^{3}}=x^4+\frac{1}{x^{4}}=2$

Hope it is clear to you.