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Werk uit m.b.v. de rekenregels

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

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Werk uit m.b.v. de rekenregels

Verbetersleutel

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