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Werk uit m.b.v. de rekenregels

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

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Werk uit m.b.v. de rekenregels

Verbetersleutel

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