411 Вправа 411Алгебра 8 клас Тарасенкова 2025 · § 10 · стор. 112

411. Обчисліть a² + a⁻², якщо:

${\displaystyle {(a+a^{-1})}^2=}$${\displaystyle a^2+2\cancel{a\cdot a^{-1}}+a^{-2}=}$${\displaystyle a^2+2+a^{-2}}$; ${\displaystyle {(a-a^{-1})}^2=}$${\displaystyle a^2-2+a^{-2}}$.

1) ${\displaystyle a+a^{-1}=}$${\displaystyle 5}$; ${\displaystyle a^2+2+a^{-2}=}$${\displaystyle 25}$; ${\displaystyle a^2+a^{-2}=}$${\displaystyle 25-2=}$${\displaystyle 23}$;

2) ${\displaystyle a-a^{-1}=}$${\displaystyle 1}$; ${\displaystyle a^2-2+a^{-2}=}$${\displaystyle 1}$; ${\displaystyle a^2+a^{-2}=}$${\displaystyle 1+2=}$${\displaystyle 3}$;

3) ${\displaystyle a+\frac{1}{a}=}$${\displaystyle 2}$; ${\displaystyle a^2+2+a^{-2}=}$${\displaystyle 4}$; ${\displaystyle a^2+a^{-2}=}$${\displaystyle 4-2=}$${\displaystyle 2}$;

4) ${\displaystyle \frac{1+a^2}{a}=}$${\displaystyle 8}$; ${\displaystyle a^{-1}+a=}$${\displaystyle 8}$; ${\displaystyle a^2+2+a^{-2}=}$${\displaystyle 64}$; ${\displaystyle a^2+a^{-2}=}$${\displaystyle 64-2=}$${\displaystyle 62}$.