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Werk uit m.b.v. de rekenregels

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

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Werk uit m.b.v. de rekenregels

Verbetersleutel

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