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Werk uit m.b.v. de rekenregels

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

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Werk uit m.b.v. de rekenregels

Verbetersleutel

1. \(\left(x^{\frac{2}{3}}\right)^{\frac{3}{4}}\\= x^{ \frac{2}{3} . \frac{3}{4} }= x^{\frac{1}{2}}\\= \sqrt{ x } \\---------------\)
2. \(\left(y^{-1}\right)^{1}\\= y^{ -1 . 1 }= y^{-1}\\=\frac{1}{y}\\---------------\)
3. \(\left(q^{\frac{5}{2}}\right)^{\frac{-2}{5}}\\= q^{ \frac{5}{2} . (\frac{-2}{5}) }= q^{-1}\\=\frac{1}{q}\\---------------\)
4. \(\left(a^{\frac{1}{3}}\right)^{\frac{-1}{6}}\\= a^{ \frac{1}{3} . (\frac{-1}{6}) }= a^{\frac{-1}{18}}\\=\frac{1}{\sqrt[18]{ a }}=\frac{1}{\sqrt[18]{ a }}. \color{purple}{\frac{\sqrt[18]{ a^{17} }}{\sqrt[18]{ a^{17} }}} \\=\frac{\sqrt[18]{ a^{17} }}{|a|}\\---------------\)
5. \(\left(y^{\frac{5}{3}}\right)^{-1}\\= y^{ \frac{5}{3} . (-1) }= y^{\frac{-5}{3}}\\=\frac{1}{\sqrt[3]{ y^{5} }}\\=\frac{1}{y.\sqrt[3]{ y^{2} }}=\frac{1}{y.\sqrt[3]{ y^{2} }} \color{purple}{\frac{\sqrt[3]{ y }}{\sqrt[3]{ y }}} \\=\frac{\sqrt[3]{ y }}{y^{2}}\\---------------\)
6. \(\left(y^{\frac{1}{5}}\right)^{\frac{-3}{5}}\\= y^{ \frac{1}{5} . (\frac{-3}{5}) }= y^{\frac{-3}{25}}\\=\frac{1}{\sqrt[25]{ y^{3} }}=\frac{1}{\sqrt[25]{ y^{3} }}. \color{purple}{\frac{\sqrt[25]{ y^{22} }}{\sqrt[25]{ y^{22} }}} \\=\frac{\sqrt[25]{ y^{22} }}{y}\\---------------\)
7. \(\left(x^{\frac{-5}{6}}\right)^{\frac{2}{3}}\\= x^{ \frac{-5}{6} . \frac{2}{3} }= x^{\frac{-5}{9}}\\=\frac{1}{\sqrt[9]{ x^{5} }}=\frac{1}{\sqrt[9]{ x^{5} }}. \color{purple}{\frac{\sqrt[9]{ x^{4} }}{\sqrt[9]{ x^{4} }}} \\=\frac{\sqrt[9]{ x^{4} }}{x}\\---------------\)
8. \(\left(y^{\frac{4}{5}}\right)^{\frac{1}{2}}\\= y^{ \frac{4}{5} . \frac{1}{2} }= y^{\frac{2}{5}}\\=\sqrt[5]{ y^{2} }\\---------------\)
9. \(\left(q^{\frac{-1}{3}}\right)^{\frac{4}{5}}\\= q^{ \frac{-1}{3} . \frac{4}{5} }= q^{\frac{-4}{15}}\\=\frac{1}{\sqrt[15]{ q^{4} }}=\frac{1}{\sqrt[15]{ q^{4} }}. \color{purple}{\frac{\sqrt[15]{ q^{11} }}{\sqrt[15]{ q^{11} }}} \\=\frac{\sqrt[15]{ q^{11} }}{q}\\---------------\)
10. \(\left(q^{\frac{1}{3}}\right)^{\frac{-3}{2}}\\= q^{ \frac{1}{3} . (\frac{-3}{2}) }= q^{\frac{-1}{2}}\\=\frac{1}{ \sqrt{ q } }=\frac{1}{ \sqrt{ q } }. \color{purple}{\frac{ \sqrt{ q } }{ \sqrt{ q } }} \\=\frac{ \sqrt{ q } }{|q|}\\---------------\)
11. \(\left(x^{\frac{-3}{2}}\right)^{\frac{3}{2}}\\= x^{ \frac{-3}{2} . \frac{3}{2} }= x^{\frac{-9}{4}}\\=\frac{1}{\sqrt[4]{ x^{9} }}\\=\frac{1}{|x^{2}|.\sqrt[4]{ x }}=\frac{1}{|x^{2}|.\sqrt[4]{ x }} \color{purple}{\frac{\sqrt[4]{ x^{3} }}{\sqrt[4]{ x^{3} }}} \\=\frac{\sqrt[4]{ x^{3} }}{|x^{3}|}\\---------------\)
12. \(\left(q^{\frac{4}{3}}\right)^{1}\\= q^{ \frac{4}{3} . 1 }= q^{\frac{4}{3}}\\=\sqrt[3]{ q^{4} }=q.\sqrt[3]{ q }\\---------------\)