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Werk uit m.b.v. de rekenregels

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

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Werk uit m.b.v. de rekenregels

Verbetersleutel

1. \(\left(a^{\frac{1}{3}}\right)^{\frac{-1}{3}}\\= a^{ \frac{1}{3} . (\frac{-1}{3}) }= a^{\frac{-1}{9}}\\=\frac{1}{\sqrt[9]{ a }}=\frac{1}{\sqrt[9]{ a }}. \color{purple}{\frac{\sqrt[9]{ a^{8} }}{\sqrt[9]{ a^{8} }}} \\=\frac{\sqrt[9]{ a^{8} }}{a}\\---------------\)
2. \(\left(x^{\frac{-4}{5}}\right)^{\frac{-5}{3}}\\= x^{ \frac{-4}{5} . (\frac{-5}{3}) }= x^{\frac{4}{3}}\\=\sqrt[3]{ x^{4} }=x.\sqrt[3]{ x }\\---------------\)
3. \(\left(x^{\frac{-1}{2}}\right)^{\frac{-4}{5}}\\= x^{ \frac{-1}{2} . (\frac{-4}{5}) }= x^{\frac{2}{5}}\\=\sqrt[5]{ x^{2} }\\---------------\)
4. \(\left(y^{\frac{-2}{3}}\right)^{\frac{-5}{6}}\\= y^{ \frac{-2}{3} . (\frac{-5}{6}) }= y^{\frac{5}{9}}\\=\sqrt[9]{ y^{5} }\\---------------\)
5. \(\left(y^{\frac{1}{4}}\right)^{\frac{-1}{6}}\\= y^{ \frac{1}{4} . (\frac{-1}{6}) }= y^{\frac{-1}{24}}\\=\frac{1}{\sqrt[24]{ y }}=\frac{1}{\sqrt[24]{ y }}. \color{purple}{\frac{\sqrt[24]{ y^{23} }}{\sqrt[24]{ y^{23} }}} \\=\frac{\sqrt[24]{ y^{23} }}{|y|}\\---------------\)
6. \(\left(a^{\frac{-5}{2}}\right)^{\frac{3}{5}}\\= a^{ \frac{-5}{2} . \frac{3}{5} }= a^{\frac{-3}{2}}\\=\frac{1}{ \sqrt{ a^{3} } }\\=\frac{1}{|a|. \sqrt{ a } }=\frac{1}{|a|. \sqrt{ a } } \color{purple}{\frac{ \sqrt{ a } }{ \sqrt{ a } }} \\=\frac{ \sqrt{ a } }{|a^{2}|}\\---------------\)
7. \(\left(q^{-1}\right)^{\frac{-1}{2}}\\= q^{ -1 . (\frac{-1}{2}) }= q^{\frac{1}{2}}\\= \sqrt{ q } \\---------------\)
8. \(\left(x^{\frac{2}{3}}\right)^{\frac{-5}{4}}\\= x^{ \frac{2}{3} . (\frac{-5}{4}) }= x^{\frac{-5}{6}}\\=\frac{1}{\sqrt[6]{ x^{5} }}=\frac{1}{\sqrt[6]{ x^{5} }}. \color{purple}{\frac{\sqrt[6]{ x }}{\sqrt[6]{ x }}} \\=\frac{\sqrt[6]{ x }}{|x|}\\---------------\)
9. \(\left(y^{\frac{-3}{4}}\right)^{\frac{5}{2}}\\= y^{ \frac{-3}{4} . \frac{5}{2} }= y^{\frac{-15}{8}}\\=\frac{1}{\sqrt[8]{ y^{15} }}\\=\frac{1}{|y|.\sqrt[8]{ y^{7} }}=\frac{1}{|y|.\sqrt[8]{ y^{7} }} \color{purple}{\frac{\sqrt[8]{ y }}{\sqrt[8]{ y }}} \\=\frac{\sqrt[8]{ y }}{|y^{2}|}\\---------------\)
10. \(\left(y^{\frac{2}{3}}\right)^{\frac{5}{2}}\\= y^{ \frac{2}{3} . \frac{5}{2} }= y^{\frac{5}{3}}\\=\sqrt[3]{ y^{5} }=y.\sqrt[3]{ y^{2} }\\---------------\)
11. \(\left(y^{\frac{1}{5}}\right)^{\frac{-4}{5}}\\= y^{ \frac{1}{5} . (\frac{-4}{5}) }= y^{\frac{-4}{25}}\\=\frac{1}{\sqrt[25]{ y^{4} }}=\frac{1}{\sqrt[25]{ y^{4} }}. \color{purple}{\frac{\sqrt[25]{ y^{21} }}{\sqrt[25]{ y^{21} }}} \\=\frac{\sqrt[25]{ y^{21} }}{y}\\---------------\)
12. \(\left(a^{\frac{-1}{3}}\right)^{-1}\\= a^{ \frac{-1}{3} . (-1) }= a^{\frac{1}{3}}\\=\sqrt[3]{ a }\\---------------\)