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Werk uit m.b.v. de rekenregels

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

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Werk uit m.b.v. de rekenregels

Verbetersleutel

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