Question

if [x+(1/x)]=20 then what is [x^2+(1/x^2)] and [x^4+(1/x^4)] ?

Answer 1

Question :

If  $\sf \red{\left(x+ \dfrac{1}{x} \right) = 20}$  then the value of  $\sf \red{(i) \left( x^2+\dfrac{1}{x^2} \right)\ and\ (ii) \left( x^4 + \dfrac{1}{x^4} \right)}$

Solution :

$\sf x+ \dfrac{1}{x} = 20$

Squaring on both sides ,

$\longrightarrow \sf \left( x+\dfrac{1}{x} \right) ^2= (20)^2$

$\bullet\ \; \sf \orange{(A+B)^2=A^2+B^2+2AB}$

$\longrightarrow \sf x^2+\dfrac{1}{x^2}+2.x.\dfrac{1}{x} = 400$

$\longrightarrow \sf x^2+\dfrac{1}{x^2}+2= 400$

$\longrightarrow \sf x^2+\dfrac{1}{x^2}= 400-2$

$\longrightarrow \sf \pink{\textsf{\textbf{x}}^{\textsf{\textbf{2}}} \textsf{\textbf{ + }} \dfrac{\textsf{\textbf{1}}}{{\textsf{\textbf{x}}}^{\textsf{\textbf{2}}}} \textsf{\textbf{ = }} \textsf{\textbf{398}}}$

Again squaring on both sides ,

$\longrightarrow \sf \left(x^2+\dfrac{1}{x^2}\right)^2= (398)^2$

$\longrightarrow \sf (x^2)^2+\dfrac{1}{(x^2)^2}+2.x^2.\dfrac{1}{x^2}= (398)^2$

$\longrightarrow \sf x^4+\dfrac{1}{x^4}+2= 158404$

$\longrightarrow \sf x^4+\dfrac{1}{x^4}= 158404-2$

$\longrightarrow \sf \purple{{\textsf{\textbf{x}}}^{\textsf{\textbf{4}}} \textsf{\textbf{ + }} \dfrac{\textsf{\textbf{1}}}{ {\textsf{\textbf{x}}}^{\textsf{\textbf{4}}}} \textsf{\textbf{ = }} \textsf{\textbf{158402}}}$

Answer 2

Given :-

If ${\bf\red{(x + \frac{1}{x} ) = 20}}$ then the value of

${\bf\red{(i) \: (x² + \frac{1}{x²} ) \: and \: (ii) \: (x⁴ + \frac{1}{x⁴}) }}$

To find :-

${\bf{x + \frac{1}{x} = 20}}$

Squaring of the both sides,

➝ ${\bf{(x + \frac{1}{x} )² = (20)²}}$

• ${\bf\pink{(A + B)² = A² + B² + 2AB}}$

➝ ${\bf{x² + \frac{1}{x²} + 2.x. \frac{1}{x} = 400}}$

➝ ${\bf{x² + \frac{1}{x²} + 2 = 400}}$

➝ ${\bf{x² + \frac{1}{x²} = 400 - 2}}$

➝ ${\bf\green{x² + \frac{1}{x²} = 398}}$

Again squaring on both sides,

➝ ${\bf{(x² + \frac{1}{x²} )² = (398)²}}$

➝ ${\bf{(x²)² + \frac{1}{(x²)²} + 2.x². \frac{1}{x²} = (398)²}}$

➝ ${\bf{x⁴ + \frac{1}{x⁴} + 2 = 158404}}$

➝ ${\bf{x⁴ + \frac{1}{x⁴} = 158404 - 2}}$

➝ ${\bf\blue{x⁴ + \frac{1}{x⁴} = 158402}}$

${\tt{Hence~Proved,}}$$\large{\bf\green{✓}}$