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Werk uit m.b.v. de rekenregels

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

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Werk uit m.b.v. de rekenregels

Verbetersleutel

1. \(\left(a^{\frac{1}{6}}\right)^{\frac{4}{3}}\\= a^{ \frac{1}{6} . \frac{4}{3} }= a^{\frac{2}{9}}\\=\sqrt[9]{ a^{2} }\\---------------\)
2. \(\left(x^{\frac{-4}{5}}\right)^{\frac{-5}{6}}\\= x^{ \frac{-4}{5} . (\frac{-5}{6}) }= x^{\frac{2}{3}}\\=\sqrt[3]{ x^{2} }\\---------------\)
3. \(\left(q^{\frac{1}{4}}\right)^{\frac{-3}{4}}\\= q^{ \frac{1}{4} . (\frac{-3}{4}) }= q^{\frac{-3}{16}}\\=\frac{1}{\sqrt[16]{ q^{3} }}=\frac{1}{\sqrt[16]{ q^{3} }}. \color{purple}{\frac{\sqrt[16]{ q^{13} }}{\sqrt[16]{ q^{13} }}} \\=\frac{\sqrt[16]{ q^{13} }}{|q|}\\---------------\)
4. \(\left(x^{\frac{-2}{3}}\right)^{\frac{5}{6}}\\= x^{ \frac{-2}{3} . \frac{5}{6} }= x^{\frac{-5}{9}}\\=\frac{1}{\sqrt[9]{ x^{5} }}=\frac{1}{\sqrt[9]{ x^{5} }}. \color{purple}{\frac{\sqrt[9]{ x^{4} }}{\sqrt[9]{ x^{4} }}} \\=\frac{\sqrt[9]{ x^{4} }}{x}\\---------------\)
5. \(\left(x^{\frac{-5}{3}}\right)^{\frac{5}{3}}\\= x^{ \frac{-5}{3} . \frac{5}{3} }= x^{\frac{-25}{9}}\\=\frac{1}{\sqrt[9]{ x^{25} }}\\=\frac{1}{x^{2}.\sqrt[9]{ x^{7} }}=\frac{1}{x^{2}.\sqrt[9]{ x^{7} }} \color{purple}{\frac{\sqrt[9]{ x^{2} }}{\sqrt[9]{ x^{2} }}} \\=\frac{\sqrt[9]{ x^{2} }}{x^{3}}\\---------------\)
6. \(\left(x^{\frac{5}{6}}\right)^{\frac{2}{5}}\\= x^{ \frac{5}{6} . \frac{2}{5} }= x^{\frac{1}{3}}\\=\sqrt[3]{ x }\\---------------\)
7. \(\left(x^{\frac{1}{4}}\right)^{\frac{3}{5}}\\= x^{ \frac{1}{4} . \frac{3}{5} }= x^{\frac{3}{20}}\\=\sqrt[20]{ x^{3} }\\---------------\)
8. \(\left(q^{\frac{-5}{6}}\right)^{\frac{5}{3}}\\= q^{ \frac{-5}{6} . \frac{5}{3} }= q^{\frac{-25}{18}}\\=\frac{1}{\sqrt[18]{ q^{25} }}\\=\frac{1}{|q|.\sqrt[18]{ q^{7} }}=\frac{1}{|q|.\sqrt[18]{ q^{7} }} \color{purple}{\frac{\sqrt[18]{ q^{11} }}{\sqrt[18]{ q^{11} }}} \\=\frac{\sqrt[18]{ q^{11} }}{|q^{2}|}\\---------------\)
9. \(\left(q^{\frac{4}{5}}\right)^{1}\\= q^{ \frac{4}{5} . 1 }= q^{\frac{4}{5}}\\=\sqrt[5]{ q^{4} }\\---------------\)
10. \(\left(a^{\frac{1}{2}}\right)^{-1}\\= a^{ \frac{1}{2} . (-1) }= a^{\frac{-1}{2}}\\=\frac{1}{ \sqrt{ a } }=\frac{1}{ \sqrt{ a } }. \color{purple}{\frac{ \sqrt{ a } }{ \sqrt{ a } }} \\=\frac{ \sqrt{ a } }{|a|}\\---------------\)
11. \(\left(y^{\frac{-5}{3}}\right)^{\frac{-4}{5}}\\= y^{ \frac{-5}{3} . (\frac{-4}{5}) }= y^{\frac{4}{3}}\\=\sqrt[3]{ y^{4} }=y.\sqrt[3]{ y }\\---------------\)
12. \(\left(y^{\frac{-5}{2}}\right)^{\frac{5}{2}}\\= y^{ \frac{-5}{2} . \frac{5}{2} }= y^{\frac{-25}{4}}\\=\frac{1}{\sqrt[4]{ y^{25} }}\\=\frac{1}{|y^{6}|.\sqrt[4]{ y }}=\frac{1}{|y^{6}|.\sqrt[4]{ y }} \color{purple}{\frac{\sqrt[4]{ y^{3} }}{\sqrt[4]{ y^{3} }}} \\=\frac{\sqrt[4]{ y^{3} }}{|y^{7}|}\\---------------\)