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Werk uit m.b.v. de rekenregels

1. \(\left(a^{\frac{-1}{3}}\right)^{\frac{5}{2}}\)
2. \(\left(x^{\frac{-1}{2}}\right)^{\frac{1}{6}}\)
3. \(\left(a^{\frac{-3}{2}}\right)^{1}\)
4. \(\left(x^{\frac{4}{3}}\right)^{\frac{-1}{6}}\)
5. \(\left(y^{\frac{-2}{3}}\right)^{\frac{5}{3}}\)
6. \(\left(a^{\frac{-1}{3}}\right)^{1}\)
7. \(\left(x^{\frac{1}{3}}\right)^{\frac{-5}{6}}\)
8. \(\left(a^{-1}\right)^{\frac{3}{5}}\)
9. \(\left(y^{\frac{-4}{3}}\right)^{-1}\)
10. \(\left(a^{\frac{-1}{2}}\right)^{\frac{-1}{2}}\)
11. \(\left(q^{\frac{2}{3}}\right)^{\frac{-3}{2}}\)
12. \(\left(y^{\frac{5}{4}}\right)^{\frac{1}{6}}\)

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Werk uit m.b.v. de rekenregels

Verbetersleutel

1. \(\left(a^{\frac{-1}{3}}\right)^{\frac{5}{2}}\\= a^{ \frac{-1}{3} . \frac{5}{2} }= a^{\frac{-5}{6}}\\=\frac{1}{\sqrt[6]{ a^{5} }}=\frac{1}{\sqrt[6]{ a^{5} }}. \color{purple}{\frac{\sqrt[6]{ a }}{\sqrt[6]{ a }}} \\=\frac{\sqrt[6]{ a }}{|a|}\\---------------\)
2. \(\left(x^{\frac{-1}{2}}\right)^{\frac{1}{6}}\\= x^{ \frac{-1}{2} . \frac{1}{6} }= x^{\frac{-1}{12}}\\=\frac{1}{\sqrt[12]{ x }}=\frac{1}{\sqrt[12]{ x }}. \color{purple}{\frac{\sqrt[12]{ x^{11} }}{\sqrt[12]{ x^{11} }}} \\=\frac{\sqrt[12]{ x^{11} }}{|x|}\\---------------\)
3. \(\left(a^{\frac{-3}{2}}\right)^{1}\\= a^{ \frac{-3}{2} . 1 }= a^{\frac{-3}{2}}\\=\frac{1}{ \sqrt{ a^{3} } }\\=\frac{1}{|a|. \sqrt{ a } }=\frac{1}{|a|. \sqrt{ a } } \color{purple}{\frac{ \sqrt{ a } }{ \sqrt{ a } }} \\=\frac{ \sqrt{ a } }{|a^{2}|}\\---------------\)
4. \(\left(x^{\frac{4}{3}}\right)^{\frac{-1}{6}}\\= x^{ \frac{4}{3} . (\frac{-1}{6}) }= x^{\frac{-2}{9}}\\=\frac{1}{\sqrt[9]{ x^{2} }}=\frac{1}{\sqrt[9]{ x^{2} }}. \color{purple}{\frac{\sqrt[9]{ x^{7} }}{\sqrt[9]{ x^{7} }}} \\=\frac{\sqrt[9]{ x^{7} }}{x}\\---------------\)
5. \(\left(y^{\frac{-2}{3}}\right)^{\frac{5}{3}}\\= y^{ \frac{-2}{3} . \frac{5}{3} }= y^{\frac{-10}{9}}\\=\frac{1}{\sqrt[9]{ y^{10} }}\\=\frac{1}{y.\sqrt[9]{ y }}=\frac{1}{y.\sqrt[9]{ y }} \color{purple}{\frac{\sqrt[9]{ y^{8} }}{\sqrt[9]{ y^{8} }}} \\=\frac{\sqrt[9]{ y^{8} }}{y^{2}}\\---------------\)
6. \(\left(a^{\frac{-1}{3}}\right)^{1}\\= a^{ \frac{-1}{3} . 1 }= a^{\frac{-1}{3}}\\=\frac{1}{\sqrt[3]{ a }}=\frac{1}{\sqrt[3]{ a }}. \color{purple}{\frac{\sqrt[3]{ a^{2} }}{\sqrt[3]{ a^{2} }}} \\=\frac{\sqrt[3]{ a^{2} }}{a}\\---------------\)
7. \(\left(x^{\frac{1}{3}}\right)^{\frac{-5}{6}}\\= x^{ \frac{1}{3} . (\frac{-5}{6}) }= x^{\frac{-5}{18}}\\=\frac{1}{\sqrt[18]{ x^{5} }}=\frac{1}{\sqrt[18]{ x^{5} }}. \color{purple}{\frac{\sqrt[18]{ x^{13} }}{\sqrt[18]{ x^{13} }}} \\=\frac{\sqrt[18]{ x^{13} }}{|x|}\\---------------\)
8. \(\left(a^{-1}\right)^{\frac{3}{5}}\\= a^{ -1 . \frac{3}{5} }= a^{\frac{-3}{5}}\\=\frac{1}{\sqrt[5]{ a^{3} }}=\frac{1}{\sqrt[5]{ a^{3} }}. \color{purple}{\frac{\sqrt[5]{ a^{2} }}{\sqrt[5]{ a^{2} }}} \\=\frac{\sqrt[5]{ a^{2} }}{a}\\---------------\)
9. \(\left(y^{\frac{-4}{3}}\right)^{-1}\\= y^{ \frac{-4}{3} . (-1) }= y^{\frac{4}{3}}\\=\sqrt[3]{ y^{4} }=y.\sqrt[3]{ y }\\---------------\)
10. \(\left(a^{\frac{-1}{2}}\right)^{\frac{-1}{2}}\\= a^{ \frac{-1}{2} . (\frac{-1}{2}) }= a^{\frac{1}{4}}\\=\sqrt[4]{ a }\\---------------\)
11. \(\left(q^{\frac{2}{3}}\right)^{\frac{-3}{2}}\\= q^{ \frac{2}{3} . (\frac{-3}{2}) }= q^{-1}\\=\frac{1}{q}\\---------------\)
12. \(\left(y^{\frac{5}{4}}\right)^{\frac{1}{6}}\\= y^{ \frac{5}{4} . \frac{1}{6} }= y^{\frac{5}{24}}\\=\sqrt[24]{ y^{5} }\\---------------\)