Задание 578

Решение:

• \({\largeа)}\ \left(\frac{2}{3}\right)^{{-}2}=\frac{1}{\left(\frac{2}{3}\right)^2}=\frac{1}{\frac{2^2}{3^2}}=\frac{3^2}{2^2}=\left(\frac{3}{2}\right)^2\)
  \(\phantom{\largeа)}\) Равенство верно.
• \({\largeб)}\ \left(\frac{1}{2}\right)^{{-}3}=\frac{1}{\left(\frac{1}{2}\right)^3}=\frac{1}{\frac{1^3}{2^3}}=\frac{2^3}{1^3}=\left(\frac{2}{1}\right)^3\)
  \(\phantom{\largeб)}\) Равенство верно.
• \({\largeв)}\ \left(\frac{12}{31}\right)^{{-}5}=\frac{1}{\left(\frac{12}{31}\right)^5}=\frac{1}{\frac{12^5}{31^5}}=\frac{31^5}{12^5}=\left(\frac{31}{12}\right)^5\)
  \(\phantom{\largeв)}\) Равенство верно.
• \({\largeг)}\ \left(\frac{3}{5}\right)^{{-}4}=\frac{1}{\left(\frac{3}{5}\right)^4}=\frac{1}{\frac{3^4}{5^4}}=\frac{5^4}{3^4}=\left(\frac{5}{3}\right)^4=\left(1\frac{2}{3}\right)^4\)
  \(\phantom{\largeг)}\) Равенство верно.