Question

if x=3+2√2,then find the value of x^2+1/x^2 and x^3+1/x^3
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Answer 1

Answer:

$\rightarrow$ $(x + \frac{1}{x} {)}^{3}=198$

$\rightarrow$ $(x + \frac{1}{x} {)}^{2} = 34$

Step-by-step explanation:

Given Information -

$x = 3 + 2 \sqrt{2}$

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To Find-

• $(x + \frac{1}{x} {)}^{3}$

• $(x + \frac{1}{x} {)}^{2}$

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Solution -

Step-1=Find the value of Reciprocal of 'x' -

• $\color{orange} {x = 3 + 2 \sqrt{2}}$

• $\color{orange} {\frac{1}{x} = \frac{1}{3 + 2 \sqrt{2} }}$

• $\color{orange} {\frac{1}{3 + 2 \sqrt{2} } \times \frac{3 - 2 \sqrt{2} }{3 - 2 \sqrt{2} }}$

• $\color{orange} {\frac{3 - 2 \sqrt{2} }{(3 + 2 \sqrt{2} )(3 - 2 \sqrt{2} )}}$

• $\color{orange} { \frac{3 - 2 \sqrt{2} }{(3 + 2 \sqrt{2} )(3 - 2 \sqrt{2} )}}$

• $\color{orange} {\frac{3 - 2 \sqrt{2} }{(3 {)}^{2} - (2 \sqrt{2} {)}^{2} }}$

• $\color{orange} {\frac{3 - 2 \sqrt{2} }{9 - 8}}$

• $\huge\sf\red {\frac{1}{x} = 3 - 2 \sqrt{2}}$

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Step-2= Find the value of $\underline{x + \frac{1}{x}}$ -

• $\color{purple} {x + \frac{1}{x}}$

• $\color{purple}{3 + 2 \sqrt{2} + 3 - 2 \sqrt{2}}$

• $\color{purple} {3 + 2 \sqrt{2} + 3 - 2 \sqrt{2}}$

• $\color{purple} {3 + 3}$

• $\huge\sf\blue{ x + \frac{1}{x}=6}$

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Step-3=Find The value of $\underline{(x + \frac{1}{x} {)}^{3}}$ -

• $\color{green}{(x + \frac{1}{x} {)}^{3} = (x + \frac{1}{x} {)}^{3} - 3(x + \frac{1}{x} )}$

• $\color{green}{{6}^{3} - (3 \times 6)}$

• $\color{green}{(6 \times 6 \times 6)- (3 \times 6)}$

• $\color{green}{216 - 18}$

• $\huge\sf\color{lime}{(x + \frac{1}{x} {)}^{3}=198}$

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Step-4=Finding $\underline{(x + \frac{1}{x} {)}^{2}}$ -

• $\color{magenta}{ (x + \frac{1}{x} {)}^{2} = (x + \frac{1}{x} {)}^{2} - 2}$

• $\color{magenta}{6² -2}$

• $\color{magenta}{36 - 2}$

• $\huge\sf\color{maroon}{(x + \frac{1}{x} {)}^{2}=34}$