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Werk uit m.b.v. de rekenregels

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

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Werk uit m.b.v. de rekenregels

Verbetersleutel

1. \(\left(q^{\frac{2}{3}}\right)^{-1}\\= q^{ \frac{2}{3} . (-1) }= q^{\frac{-2}{3}}\\=\frac{1}{\sqrt[3]{ q^{2} }}=\frac{1}{\sqrt[3]{ q^{2} }}. \color{purple}{\frac{\sqrt[3]{ q }}{\sqrt[3]{ q }}} \\=\frac{\sqrt[3]{ q }}{q}\\---------------\)
2. \(\left(x^{\frac{1}{6}}\right)^{\frac{1}{5}}\\= x^{ \frac{1}{6} . \frac{1}{5} }= x^{\frac{1}{30}}\\=\sqrt[30]{ x }\\---------------\)
3. \(\left(y^{\frac{-5}{3}}\right)^{\frac{-5}{3}}\\= y^{ \frac{-5}{3} . (\frac{-5}{3}) }= y^{\frac{25}{9}}\\=\sqrt[9]{ y^{25} }=y^{2}.\sqrt[9]{ y^{7} }\\---------------\)
4. \(\left(y^{\frac{4}{3}}\right)^{-2}\\= y^{ \frac{4}{3} . (-2) }= y^{\frac{-8}{3}}\\=\frac{1}{\sqrt[3]{ y^{8} }}\\=\frac{1}{y^{2}.\sqrt[3]{ y^{2} }}=\frac{1}{y^{2}.\sqrt[3]{ y^{2} }} \color{purple}{\frac{\sqrt[3]{ y }}{\sqrt[3]{ y }}} \\=\frac{\sqrt[3]{ y }}{y^{3}}\\---------------\)
5. \(\left(x^{\frac{1}{3}}\right)^{\frac{-5}{4}}\\= x^{ \frac{1}{3} . (\frac{-5}{4}) }= x^{\frac{-5}{12}}\\=\frac{1}{\sqrt[12]{ x^{5} }}=\frac{1}{\sqrt[12]{ x^{5} }}. \color{purple}{\frac{\sqrt[12]{ x^{7} }}{\sqrt[12]{ x^{7} }}} \\=\frac{\sqrt[12]{ x^{7} }}{|x|}\\---------------\)
6. \(\left(x^{\frac{-1}{2}}\right)^{\frac{-1}{2}}\\= x^{ \frac{-1}{2} . (\frac{-1}{2}) }= x^{\frac{1}{4}}\\=\sqrt[4]{ x }\\---------------\)
7. \(\left(x^{1}\right)^{-1}\\= x^{ 1 . (-1) }= x^{-1}\\=\frac{1}{x}\\---------------\)
8. \(\left(q^{\frac{-4}{5}}\right)^{1}\\= q^{ \frac{-4}{5} . 1 }= q^{\frac{-4}{5}}\\=\frac{1}{\sqrt[5]{ q^{4} }}=\frac{1}{\sqrt[5]{ q^{4} }}. \color{purple}{\frac{\sqrt[5]{ q }}{\sqrt[5]{ q }}} \\=\frac{\sqrt[5]{ q }}{q}\\---------------\)
9. \(\left(x^{-1}\right)^{\frac{1}{3}}\\= x^{ -1 . \frac{1}{3} }= x^{\frac{-1}{3}}\\=\frac{1}{\sqrt[3]{ x }}=\frac{1}{\sqrt[3]{ x }}. \color{purple}{\frac{\sqrt[3]{ x^{2} }}{\sqrt[3]{ x^{2} }}} \\=\frac{\sqrt[3]{ x^{2} }}{x}\\---------------\)
10. \(\left(x^{\frac{5}{6}}\right)^{\frac{-5}{3}}\\= x^{ \frac{5}{6} . (\frac{-5}{3}) }= x^{\frac{-25}{18}}\\=\frac{1}{\sqrt[18]{ x^{25} }}\\=\frac{1}{|x|.\sqrt[18]{ x^{7} }}=\frac{1}{|x|.\sqrt[18]{ x^{7} }} \color{purple}{\frac{\sqrt[18]{ x^{11} }}{\sqrt[18]{ x^{11} }}} \\=\frac{\sqrt[18]{ x^{11} }}{|x^{2}|}\\---------------\)
11. \(\left(x^{1}\right)^{\frac{-2}{5}}\\= x^{ 1 . (\frac{-2}{5}) }= x^{\frac{-2}{5}}\\=\frac{1}{\sqrt[5]{ x^{2} }}=\frac{1}{\sqrt[5]{ x^{2} }}. \color{purple}{\frac{\sqrt[5]{ x^{3} }}{\sqrt[5]{ x^{3} }}} \\=\frac{\sqrt[5]{ x^{3} }}{x}\\---------------\)
12. \(\left(y^{-1}\right)^{2}\\= y^{ -1 . 2 }= y^{-2}\\=\frac{1}{y^{2}}\\---------------\)