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Werk uit m.b.v. de rekenregels

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

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Werk uit m.b.v. de rekenregels

Verbetersleutel

1. \(\left(x^{\frac{1}{3}}\right)^{\frac{-1}{3}}\\= x^{ \frac{1}{3} . (\frac{-1}{3}) }= x^{\frac{-1}{9}}\\=\frac{1}{\sqrt[9]{ x }}=\frac{1}{\sqrt[9]{ x }}. \color{purple}{\frac{\sqrt[9]{ x^{8} }}{\sqrt[9]{ x^{8} }}} \\=\frac{\sqrt[9]{ x^{8} }}{x}\\---------------\)
2. \(\left(a^{\frac{3}{4}}\right)^{\frac{1}{5}}\\= a^{ \frac{3}{4} . \frac{1}{5} }= a^{\frac{3}{20}}\\=\sqrt[20]{ a^{3} }\\---------------\)
3. \(\left(x^{\frac{1}{6}}\right)^{\frac{-5}{4}}\\= x^{ \frac{1}{6} . (\frac{-5}{4}) }= x^{\frac{-5}{24}}\\=\frac{1}{\sqrt[24]{ x^{5} }}=\frac{1}{\sqrt[24]{ x^{5} }}. \color{purple}{\frac{\sqrt[24]{ x^{19} }}{\sqrt[24]{ x^{19} }}} \\=\frac{\sqrt[24]{ x^{19} }}{|x|}\\---------------\)
4. \(\left(x^{\frac{-3}{5}}\right)^{\frac{-1}{4}}\\= x^{ \frac{-3}{5} . (\frac{-1}{4}) }= x^{\frac{3}{20}}\\=\sqrt[20]{ x^{3} }\\---------------\)
5. \(\left(y^{\frac{-1}{4}}\right)^{\frac{-5}{2}}\\= y^{ \frac{-1}{4} . (\frac{-5}{2}) }= y^{\frac{5}{8}}\\=\sqrt[8]{ y^{5} }\\---------------\)
6. \(\left(x^{\frac{-5}{3}}\right)^{\frac{2}{3}}\\= x^{ \frac{-5}{3} . \frac{2}{3} }= x^{\frac{-10}{9}}\\=\frac{1}{\sqrt[9]{ x^{10} }}\\=\frac{1}{x.\sqrt[9]{ x }}=\frac{1}{x.\sqrt[9]{ x }} \color{purple}{\frac{\sqrt[9]{ x^{8} }}{\sqrt[9]{ x^{8} }}} \\=\frac{\sqrt[9]{ x^{8} }}{x^{2}}\\---------------\)
7. \(\left(y^{\frac{1}{3}}\right)^{\frac{-5}{6}}\\= y^{ \frac{1}{3} . (\frac{-5}{6}) }= y^{\frac{-5}{18}}\\=\frac{1}{\sqrt[18]{ y^{5} }}=\frac{1}{\sqrt[18]{ y^{5} }}. \color{purple}{\frac{\sqrt[18]{ y^{13} }}{\sqrt[18]{ y^{13} }}} \\=\frac{\sqrt[18]{ y^{13} }}{|y|}\\---------------\)
8. \(\left(a^{-1}\right)^{1}\\= a^{ -1 . 1 }= a^{-1}\\=\frac{1}{a}\\---------------\)
9. \(\left(a^{\frac{-2}{3}}\right)^{\frac{3}{4}}\\= a^{ \frac{-2}{3} . \frac{3}{4} }= a^{\frac{-1}{2}}\\=\frac{1}{ \sqrt{ a } }=\frac{1}{ \sqrt{ a } }. \color{purple}{\frac{ \sqrt{ a } }{ \sqrt{ a } }} \\=\frac{ \sqrt{ a } }{|a|}\\---------------\)
10. \(\left(a^{1}\right)^{\frac{-1}{5}}\\= a^{ 1 . (\frac{-1}{5}) }= a^{\frac{-1}{5}}\\=\frac{1}{\sqrt[5]{ a }}=\frac{1}{\sqrt[5]{ a }}. \color{purple}{\frac{\sqrt[5]{ a^{4} }}{\sqrt[5]{ a^{4} }}} \\=\frac{\sqrt[5]{ a^{4} }}{a}\\---------------\)
11. \(\left(a^{\frac{2}{3}}\right)^{\frac{1}{2}}\\= a^{ \frac{2}{3} . \frac{1}{2} }= a^{\frac{1}{3}}\\=\sqrt[3]{ a }\\---------------\)
12. \(\left(a^{\frac{1}{2}}\right)^{\frac{-5}{2}}\\= a^{ \frac{1}{2} . (\frac{-5}{2}) }= a^{\frac{-5}{4}}\\=\frac{1}{\sqrt[4]{ a^{5} }}\\=\frac{1}{|a|.\sqrt[4]{ a }}=\frac{1}{|a|.\sqrt[4]{ a }} \color{purple}{\frac{\sqrt[4]{ a^{3} }}{\sqrt[4]{ a^{3} }}} \\=\frac{\sqrt[4]{ a^{3} }}{|a^{2}|}\\---------------\)