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Werk uit m.b.v. de rekenregels

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

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Werk uit m.b.v. de rekenregels

Verbetersleutel

1. \(\left(y^{\frac{-1}{3}}\right)^{\frac{5}{6}}\\= y^{ \frac{-1}{3} . \frac{5}{6} }= y^{\frac{-5}{18}}\\=\frac{1}{\sqrt[18]{ y^{5} }}=\frac{1}{\sqrt[18]{ y^{5} }}. \color{purple}{\frac{\sqrt[18]{ y^{13} }}{\sqrt[18]{ y^{13} }}} \\=\frac{\sqrt[18]{ y^{13} }}{|y|}\\---------------\)
2. \(\left(y^{\frac{-1}{2}}\right)^{\frac{1}{2}}\\= y^{ \frac{-1}{2} . \frac{1}{2} }= y^{\frac{-1}{4}}\\=\frac{1}{\sqrt[4]{ y }}=\frac{1}{\sqrt[4]{ y }}. \color{purple}{\frac{\sqrt[4]{ y^{3} }}{\sqrt[4]{ y^{3} }}} \\=\frac{\sqrt[4]{ y^{3} }}{|y|}\\---------------\)
3. \(\left(q^{\frac{5}{2}}\right)^{-1}\\= q^{ \frac{5}{2} . (-1) }= q^{\frac{-5}{2}}\\=\frac{1}{ \sqrt{ q^{5} } }\\=\frac{1}{|q^{2}|. \sqrt{ q } }=\frac{1}{|q^{2}|. \sqrt{ q } } \color{purple}{\frac{ \sqrt{ q } }{ \sqrt{ q } }} \\=\frac{ \sqrt{ q } }{|q^{3}|}\\---------------\)
4. \(\left(y^{\frac{5}{4}}\right)^{\frac{-1}{2}}\\= y^{ \frac{5}{4} . (\frac{-1}{2}) }= y^{\frac{-5}{8}}\\=\frac{1}{\sqrt[8]{ y^{5} }}=\frac{1}{\sqrt[8]{ y^{5} }}. \color{purple}{\frac{\sqrt[8]{ y^{3} }}{\sqrt[8]{ y^{3} }}} \\=\frac{\sqrt[8]{ y^{3} }}{|y|}\\---------------\)
5. \(\left(q^{1}\right)^{\frac{-4}{3}}\\= q^{ 1 . (\frac{-4}{3}) }= q^{\frac{-4}{3}}\\=\frac{1}{\sqrt[3]{ q^{4} }}\\=\frac{1}{q.\sqrt[3]{ q }}=\frac{1}{q.\sqrt[3]{ q }} \color{purple}{\frac{\sqrt[3]{ q^{2} }}{\sqrt[3]{ q^{2} }}} \\=\frac{\sqrt[3]{ q^{2} }}{q^{2}}\\---------------\)
6. \(\left(x^{\frac{-5}{4}}\right)^{\frac{4}{5}}\\= x^{ \frac{-5}{4} . \frac{4}{5} }= x^{-1}\\=\frac{1}{x}\\---------------\)
7. \(\left(q^{-1}\right)^{\frac{1}{6}}\\= q^{ -1 . \frac{1}{6} }= q^{\frac{-1}{6}}\\=\frac{1}{\sqrt[6]{ q }}=\frac{1}{\sqrt[6]{ q }}. \color{purple}{\frac{\sqrt[6]{ q^{5} }}{\sqrt[6]{ q^{5} }}} \\=\frac{\sqrt[6]{ q^{5} }}{|q|}\\---------------\)
8. \(\left(x^{1}\right)^{\frac{2}{3}}\\= x^{ 1 . \frac{2}{3} }= x^{\frac{2}{3}}\\=\sqrt[3]{ x^{2} }\\---------------\)
9. \(\left(y^{\frac{-1}{3}}\right)^{\frac{-2}{5}}\\= y^{ \frac{-1}{3} . (\frac{-2}{5}) }= y^{\frac{2}{15}}\\=\sqrt[15]{ y^{2} }\\---------------\)
10. \(\left(q^{\frac{1}{3}}\right)^{1}\\= q^{ \frac{1}{3} . 1 }= q^{\frac{1}{3}}\\=\sqrt[3]{ q }\\---------------\)
11. \(\left(x^{2}\right)^{\frac{-1}{4}}\\= x^{ 2 . (\frac{-1}{4}) }= x^{\frac{-1}{2}}\\=\frac{1}{ \sqrt{ x } }=\frac{1}{ \sqrt{ x } }. \color{purple}{\frac{ \sqrt{ x } }{ \sqrt{ x } }} \\=\frac{ \sqrt{ x } }{|x|}\\---------------\)
12. \(\left(y^{\frac{-3}{4}}\right)^{\frac{5}{6}}\\= y^{ \frac{-3}{4} . \frac{5}{6} }= y^{\frac{-5}{8}}\\=\frac{1}{\sqrt[8]{ y^{5} }}=\frac{1}{\sqrt[8]{ y^{5} }}. \color{purple}{\frac{\sqrt[8]{ y^{3} }}{\sqrt[8]{ y^{3} }}} \\=\frac{\sqrt[8]{ y^{3} }}{|y|}\\---------------\)