Zadanie 5.5
a)
a^{\frac{1}{2}}*a^{\frac{1}{4}}=a^{\frac{2}{4}+\frac{1}{4}}=a^{\frac{3}{4}}
b^{\frac{1}{2}}*b^{\frac{1}{3}}=b^{\frac{3}{6}+\frac{2}{6}}=b^{\frac{5}{6}}
c^{\frac{1}{2}}*c^{-\frac{1}{4}}=c^{\frac{2}{4}-\frac{1}{4}}=c^{\frac{1}{4}} = \sqrt[4]{c}
x^{\frac{1}{2}}*x^{-\frac{1}{6}}=x^{\frac{3}{6}-\frac{1}{6}}=x^{\frac{2}{6}}=x^{\frac{1}{3}} = \sqrt[3]{x}
b)
a*a^{-\frac{2}{3}}=a^{1-\frac{2}{3}}=a^{\frac{1}{3}} = \sqrt[3]a
x^{\frac{1}{2}}\sqrt{x}=\sqrt x*\sqrt x=x
x^{\frac{2}{3}}\sqrt[3]{x}=x^{\frac{2}{3}}* x^{\frac{1}{3}}=x^{\frac{2}{3}+\frac{1}{3}}=x
x^{\frac{1}{5}}\sqrt[3]{x^3} = x^{\frac{1}{5}}*x=x^{\frac{1}{5}+\frac{5}{5}}=x^{\frac{6}{5}}
c)
a^{\frac{1}{m}}*a^{\frac{1}{n}}*a^{\frac{1}{r}}=a^{\frac{1}{m}+\frac{1}{n}+\frac{1}{r}}
\frac{1}{x^{a+b}}*\frac{1}{x^{a-b}}*x^{\frac{-2a}{a^2-b^2}}=\frac{x^{\frac{-2a}{a^2-b^2}}}{x^{a+b+a-b}}=\frac{x^{\frac{-2a}{a^2-b^2}}}{x^{2a}}=x^{\frac{-2a}{a^2-b^2}-2a}