Question

if x +1/x=5, find the values of
a. x^2+1/x^2
b. x^4+1/4^4​

Answer 1

${\bold{\tt{⟹x+\frac{1}{x}=5}}}$

$\red{\boxed{\tt{(x+\frac{1}{x})²=x²+\frac{1}{x²}+2(x)(\frac{1}{x})}}}$

${\bold{\tt{⟹x²+\frac{1}{x²}=(x+\frac{1}{x})²-2(x)(\frac{1}{x})}}}$

${\bold{\tt{⟹x²+\frac{1}{x²}=(5)²-2(1)}}}$

$⟹$$\red{\bold{\tt{x²+\frac{1}{x²}=23}}}$

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${\bold{\tt{⟹x²+\frac{1}{x²}=23}}}$

$\red{\boxed{\tt{(x²+\frac{1}{x²})²=x⁴+\frac{1}{x⁴}+2(x²)(\frac{1}{x²})}}}$

${\bold{\tt{⟹x⁴+\frac{1}{x⁴}=(x²+\frac{1}{x²})²-2(x²)(\frac{1}{x²})}}}$

${\bold{\tt{⟹x⁴+\frac{1}{x⁴}=(23)²-2(1)}}}$

$⟹$$\red{\bold{\tt{x⁴+\frac{1}{x⁴}=527}}}$

Answer 2

$\bigstar$ EXPLAINATION $\bigstar$

$x +\frac{1}{x} = 5$

Squaring on both sides

$( x + \frac{1}{x} )^2 = 25$

$x^2 + \frac{1}{x^2} + \frac{2x}{x} = 25$

$x^2 + \frac{1}{x^2} + 2 = 25$

$x^2 + \frac{1}{x^2} = 23$

Squaring on both sides

$(x^2 + \frac{1}{x^2})^2 = 23^2$

$x^4 + \frac{1}{x^4}+\frac{2x^2}{x^2} = 529$

$x^4 + \frac{1}{x^4} + 2 =529$

$x^4 + \frac{1}{x^4} = 527$