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Werk uit m.b.v. de rekenregels

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

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Werk uit m.b.v. de rekenregels

Verbetersleutel

1. \(\left(x^{\frac{-5}{3}}\right)^{\frac{5}{6}}\\= x^{ \frac{-5}{3} . \frac{5}{6} }= x^{\frac{-25}{18}}\\=\frac{1}{\sqrt[18]{ x^{25} }}\\=\frac{1}{|x|.\sqrt[18]{ x^{7} }}=\frac{1}{|x|.\sqrt[18]{ x^{7} }} \color{purple}{\frac{\sqrt[18]{ x^{11} }}{\sqrt[18]{ x^{11} }}} \\=\frac{\sqrt[18]{ x^{11} }}{|x^{2}|}\\---------------\)
2. \(\left(y^{\frac{1}{4}}\right)^{\frac{-2}{3}}\\= y^{ \frac{1}{4} . (\frac{-2}{3}) }= y^{\frac{-1}{6}}\\=\frac{1}{\sqrt[6]{ y }}=\frac{1}{\sqrt[6]{ y }}. \color{purple}{\frac{\sqrt[6]{ y^{5} }}{\sqrt[6]{ y^{5} }}} \\=\frac{\sqrt[6]{ y^{5} }}{|y|}\\---------------\)
3. \(\left(y^{\frac{-1}{5}}\right)^{\frac{1}{2}}\\= y^{ \frac{-1}{5} . \frac{1}{2} }= y^{\frac{-1}{10}}\\=\frac{1}{\sqrt[10]{ y }}=\frac{1}{\sqrt[10]{ y }}. \color{purple}{\frac{\sqrt[10]{ y^{9} }}{\sqrt[10]{ y^{9} }}} \\=\frac{\sqrt[10]{ y^{9} }}{|y|}\\---------------\)
4. \(\left(y^{\frac{2}{3}}\right)^{\frac{2}{3}}\\= y^{ \frac{2}{3} . \frac{2}{3} }= y^{\frac{4}{9}}\\=\sqrt[9]{ y^{4} }\\---------------\)
5. \(\left(x^{\frac{-5}{4}}\right)^{\frac{-4}{3}}\\= x^{ \frac{-5}{4} . (\frac{-4}{3}) }= x^{\frac{5}{3}}\\=\sqrt[3]{ x^{5} }=x.\sqrt[3]{ x^{2} }\\---------------\)
6. \(\left(y^{\frac{3}{2}}\right)^{\frac{1}{2}}\\= y^{ \frac{3}{2} . \frac{1}{2} }= y^{\frac{3}{4}}\\=\sqrt[4]{ y^{3} }\\---------------\)
7. \(\left(q^{-1}\right)^{\frac{-1}{5}}\\= q^{ -1 . (\frac{-1}{5}) }= q^{\frac{1}{5}}\\=\sqrt[5]{ q }\\---------------\)
8. \(\left(a^{\frac{1}{4}}\right)^{\frac{-1}{5}}\\= a^{ \frac{1}{4} . (\frac{-1}{5}) }= a^{\frac{-1}{20}}\\=\frac{1}{\sqrt[20]{ a }}=\frac{1}{\sqrt[20]{ a }}. \color{purple}{\frac{\sqrt[20]{ a^{19} }}{\sqrt[20]{ a^{19} }}} \\=\frac{\sqrt[20]{ a^{19} }}{|a|}\\---------------\)
9. \(\left(a^{\frac{5}{2}}\right)^{-1}\\= a^{ \frac{5}{2} . (-1) }= a^{\frac{-5}{2}}\\=\frac{1}{ \sqrt{ a^{5} } }\\=\frac{1}{|a^{2}|. \sqrt{ a } }=\frac{1}{|a^{2}|. \sqrt{ a } } \color{purple}{\frac{ \sqrt{ a } }{ \sqrt{ a } }} \\=\frac{ \sqrt{ a } }{|a^{3}|}\\---------------\)
10. \(\left(x^{\frac{-2}{5}}\right)^{-1}\\= x^{ \frac{-2}{5} . (-1) }= x^{\frac{2}{5}}\\=\sqrt[5]{ x^{2} }\\---------------\)
11. \(\left(q^{\frac{1}{2}}\right)^{\frac{-5}{4}}\\= q^{ \frac{1}{2} . (\frac{-5}{4}) }= q^{\frac{-5}{8}}\\=\frac{1}{\sqrt[8]{ q^{5} }}=\frac{1}{\sqrt[8]{ q^{5} }}. \color{purple}{\frac{\sqrt[8]{ q^{3} }}{\sqrt[8]{ q^{3} }}} \\=\frac{\sqrt[8]{ q^{3} }}{|q|}\\---------------\)
12. \(\left(a^{-1}\right)^{\frac{1}{2}}\\= a^{ -1 . \frac{1}{2} }= a^{\frac{-1}{2}}\\=\frac{1}{ \sqrt{ a } }=\frac{1}{ \sqrt{ a } }. \color{purple}{\frac{ \sqrt{ a } }{ \sqrt{ a } }} \\=\frac{ \sqrt{ a } }{|a|}\\---------------\)