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Werk uit m.b.v. de rekenregels

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

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Werk uit m.b.v. de rekenregels

Verbetersleutel

1. \(\left(y^{\frac{-2}{3}}\right)^{\frac{-4}{3}}\\= y^{ \frac{-2}{3} . (\frac{-4}{3}) }= y^{\frac{8}{9}}\\=\sqrt[9]{ y^{8} }\\---------------\)
2. \(\left(y^{\frac{5}{2}}\right)^{-1}\\= y^{ \frac{5}{2} . (-1) }= y^{\frac{-5}{2}}\\=\frac{1}{ \sqrt{ y^{5} } }\\=\frac{1}{|y^{2}|. \sqrt{ y } }=\frac{1}{|y^{2}|. \sqrt{ y } } \color{purple}{\frac{ \sqrt{ y } }{ \sqrt{ y } }} \\=\frac{ \sqrt{ y } }{|y^{3}|}\\---------------\)
3. \(\left(q^{\frac{2}{3}}\right)^{\frac{2}{5}}\\= q^{ \frac{2}{3} . \frac{2}{5} }= q^{\frac{4}{15}}\\=\sqrt[15]{ q^{4} }\\---------------\)
4. \(\left(y^{1}\right)^{\frac{1}{4}}\\= y^{ 1 . \frac{1}{4} }= y^{\frac{1}{4}}\\=\sqrt[4]{ y }\\---------------\)
5. \(\left(a^{\frac{2}{3}}\right)^{-1}\\= a^{ \frac{2}{3} . (-1) }= a^{\frac{-2}{3}}\\=\frac{1}{\sqrt[3]{ a^{2} }}=\frac{1}{\sqrt[3]{ a^{2} }}. \color{purple}{\frac{\sqrt[3]{ a }}{\sqrt[3]{ a }}} \\=\frac{\sqrt[3]{ a }}{a}\\---------------\)
6. \(\left(q^{-1}\right)^{\frac{3}{2}}\\= q^{ -1 . \frac{3}{2} }= q^{\frac{-3}{2}}\\=\frac{1}{ \sqrt{ q^{3} } }\\=\frac{1}{|q|. \sqrt{ q } }=\frac{1}{|q|. \sqrt{ q } } \color{purple}{\frac{ \sqrt{ q } }{ \sqrt{ q } }} \\=\frac{ \sqrt{ q } }{|q^{2}|}\\---------------\)
7. \(\left(a^{\frac{1}{4}}\right)^{\frac{-5}{6}}\\= a^{ \frac{1}{4} . (\frac{-5}{6}) }= a^{\frac{-5}{24}}\\=\frac{1}{\sqrt[24]{ a^{5} }}=\frac{1}{\sqrt[24]{ a^{5} }}. \color{purple}{\frac{\sqrt[24]{ a^{19} }}{\sqrt[24]{ a^{19} }}} \\=\frac{\sqrt[24]{ a^{19} }}{|a|}\\---------------\)
8. \(\left(x^{2}\right)^{\frac{1}{3}}\\= x^{ 2 . \frac{1}{3} }= x^{\frac{2}{3}}\\=\sqrt[3]{ x^{2} }\\---------------\)
9. \(\left(y^{\frac{1}{4}}\right)^{\frac{-3}{4}}\\= y^{ \frac{1}{4} . (\frac{-3}{4}) }= y^{\frac{-3}{16}}\\=\frac{1}{\sqrt[16]{ y^{3} }}=\frac{1}{\sqrt[16]{ y^{3} }}. \color{purple}{\frac{\sqrt[16]{ y^{13} }}{\sqrt[16]{ y^{13} }}} \\=\frac{\sqrt[16]{ y^{13} }}{|y|}\\---------------\)
10. \(\left(y^{\frac{-1}{2}}\right)^{\frac{2}{5}}\\= y^{ \frac{-1}{2} . \frac{2}{5} }= y^{\frac{-1}{5}}\\=\frac{1}{\sqrt[5]{ y }}=\frac{1}{\sqrt[5]{ y }}. \color{purple}{\frac{\sqrt[5]{ y^{4} }}{\sqrt[5]{ y^{4} }}} \\=\frac{\sqrt[5]{ y^{4} }}{y}\\---------------\)
11. \(\left(a^{\frac{-2}{3}}\right)^{\frac{1}{5}}\\= a^{ \frac{-2}{3} . \frac{1}{5} }= a^{\frac{-2}{15}}\\=\frac{1}{\sqrt[15]{ a^{2} }}=\frac{1}{\sqrt[15]{ a^{2} }}. \color{purple}{\frac{\sqrt[15]{ a^{13} }}{\sqrt[15]{ a^{13} }}} \\=\frac{\sqrt[15]{ a^{13} }}{a}\\---------------\)
12. \(\left(q^{\frac{3}{4}}\right)^{\frac{-5}{3}}\\= q^{ \frac{3}{4} . (\frac{-5}{3}) }= q^{\frac{-5}{4}}\\=\frac{1}{\sqrt[4]{ q^{5} }}\\=\frac{1}{|q|.\sqrt[4]{ q }}=\frac{1}{|q|.\sqrt[4]{ q }} \color{purple}{\frac{\sqrt[4]{ q^{3} }}{\sqrt[4]{ q^{3} }}} \\=\frac{\sqrt[4]{ q^{3} }}{|q^{2}|}\\---------------\)