Question

Dado que x+1x=5 obtenha o valor de: a) x²+1/x²
b) x4+1/x4 ( O quatro está elevado)

Answer 1

Seja: $x+ \frac{1}{x}=5$, determine:
a) $x^2+ \frac{1}{x^2}$
b) $x^4+ \frac{1}{x^4}$

a) 
$x+ \frac{1}{x}=5\\ \\\left(x+ \frac{1}{x}\right)^2=5^2\\ \\x^2+2\cdot x \cdot \frac{1}{x}+ \frac{1}{x^2}=25 \\ \\x^2+2+ \frac{1}{x^2}=25 \\ \\x^2+ \frac{1}{x^2}=25 -2\\ \\x^2+ \frac{1}{x^2}=23$

b)
$\\x^2+ \frac{1}{x^2}=23\\ \\\left(x^2+ \frac{1}{x^2}\right)^2=23^2\\ \\x^4+2 \cdot x^2 \cdot \frac{1}{x^2} + \frac{1}{x^4} =529\\ \\x^4+2 + \frac{1}{x^4} =529\\ \\x^4 + \frac{1}{x^4} =529-2\\ \\x^4 + \frac{1}{x^4} =527$