Question

sabendo que (a + 1/a =2), então calcule a) a² + 1/a² b) a³ + 1/a³ ​

Answer 1

$\displaystyle \sf a+\frac{1}{a} = 2 \ \ ;\ \ a^2+\frac{1}{a^2} = \ ? \ \ ; \ \ a^3+\frac{1}{a^3} = \ ? \\\\ \text{Fa{\c c}amos } : \\\\ \left (a+\frac{1}{a} \right) ^2 = 2^2 \\\\\\ a^2+2 \cdot a\cdot \frac{1}{a} + \frac{1}{a^2} = 4 \\\\\\ a^2+\frac{1}{a^2} = 4-2 \\\\\\ \huge\boxed{\ \sf a^2+\frac{1}{a^2} = 2 \ }\checkmark$

$\displaystyle \sf \text{Fa{\c c}amos }: \\\\ \left (a+\frac{1}{a} \right)^3 = 2^3 \ \ \ ; \ \ \boxed{\sf obs : (a+b)^3=a^3+b^3+3\cdot a\cdot b\cdot (a+b)}\\\\\\ a^3+\frac{1}{a^3} + 3\cdot a \cdot \frac{1}{a}\cdot \underbrace{\sf \left(a+\frac{1}{a} \right) }_{2} = 8 \\\\\\\ a^3+\frac{1}{a^3}+3\cdot 2 = 8 \\\\\\ a^3+\frac{1}{a^3} = 8-6 \\\\\\ \huge\boxed{\sf a^3+\frac{1}{a^3} = 2\ } \checkmark$