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Werk uit m.b.v. de rekenregels

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

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Werk uit m.b.v. de rekenregels

Verbetersleutel

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