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Werk uit m.b.v. de rekenregels

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

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Werk uit m.b.v. de rekenregels

Verbetersleutel

1. \(\left(a^{\frac{5}{4}}\right)^{1}\\= a^{ \frac{5}{4} . 1 }= a^{\frac{5}{4}}\\=\sqrt[4]{ a^{5} }=|a|.\sqrt[4]{ a }\\---------------\)
2. \(\left(a^{\frac{-1}{4}}\right)^{\frac{1}{3}}\\= a^{ \frac{-1}{4} . \frac{1}{3} }= a^{\frac{-1}{12}}\\=\frac{1}{\sqrt[12]{ a }}=\frac{1}{\sqrt[12]{ a }}. \color{purple}{\frac{\sqrt[12]{ a^{11} }}{\sqrt[12]{ a^{11} }}} \\=\frac{\sqrt[12]{ a^{11} }}{|a|}\\---------------\)
3. \(\left(q^{1}\right)^{\frac{2}{5}}\\= q^{ 1 . \frac{2}{5} }= q^{\frac{2}{5}}\\=\sqrt[5]{ q^{2} }\\---------------\)
4. \(\left(a^{\frac{2}{3}}\right)^{\frac{-2}{3}}\\= a^{ \frac{2}{3} . (\frac{-2}{3}) }= a^{\frac{-4}{9}}\\=\frac{1}{\sqrt[9]{ a^{4} }}=\frac{1}{\sqrt[9]{ a^{4} }}. \color{purple}{\frac{\sqrt[9]{ a^{5} }}{\sqrt[9]{ a^{5} }}} \\=\frac{\sqrt[9]{ a^{5} }}{a}\\---------------\)
5. \(\left(y^{\frac{1}{3}}\right)^{-1}\\= y^{ \frac{1}{3} . (-1) }= y^{\frac{-1}{3}}\\=\frac{1}{\sqrt[3]{ y }}=\frac{1}{\sqrt[3]{ y }}. \color{purple}{\frac{\sqrt[3]{ y^{2} }}{\sqrt[3]{ y^{2} }}} \\=\frac{\sqrt[3]{ y^{2} }}{y}\\---------------\)
6. \(\left(a^{\frac{-2}{3}}\right)^{\frac{4}{5}}\\= a^{ \frac{-2}{3} . \frac{4}{5} }= a^{\frac{-8}{15}}\\=\frac{1}{\sqrt[15]{ a^{8} }}=\frac{1}{\sqrt[15]{ a^{8} }}. \color{purple}{\frac{\sqrt[15]{ a^{7} }}{\sqrt[15]{ a^{7} }}} \\=\frac{\sqrt[15]{ a^{7} }}{a}\\---------------\)
7. \(\left(q^{\frac{-5}{6}}\right)^{\frac{-4}{5}}\\= q^{ \frac{-5}{6} . (\frac{-4}{5}) }= q^{\frac{2}{3}}\\=\sqrt[3]{ q^{2} }\\---------------\)
8. \(\left(x^{\frac{5}{4}}\right)^{\frac{-5}{6}}\\= x^{ \frac{5}{4} . (\frac{-5}{6}) }= x^{\frac{-25}{24}}\\=\frac{1}{\sqrt[24]{ x^{25} }}\\=\frac{1}{|x|.\sqrt[24]{ x }}=\frac{1}{|x|.\sqrt[24]{ x }} \color{purple}{\frac{\sqrt[24]{ x^{23} }}{\sqrt[24]{ x^{23} }}} \\=\frac{\sqrt[24]{ x^{23} }}{|x^{2}|}\\---------------\)
9. \(\left(q^{\frac{1}{2}}\right)^{-1}\\= q^{ \frac{1}{2} . (-1) }= q^{\frac{-1}{2}}\\=\frac{1}{ \sqrt{ q } }=\frac{1}{ \sqrt{ q } }. \color{purple}{\frac{ \sqrt{ q } }{ \sqrt{ q } }} \\=\frac{ \sqrt{ q } }{|q|}\\---------------\)
10. \(\left(y^{\frac{1}{3}}\right)^{\frac{-5}{6}}\\= y^{ \frac{1}{3} . (\frac{-5}{6}) }= y^{\frac{-5}{18}}\\=\frac{1}{\sqrt[18]{ y^{5} }}=\frac{1}{\sqrt[18]{ y^{5} }}. \color{purple}{\frac{\sqrt[18]{ y^{13} }}{\sqrt[18]{ y^{13} }}} \\=\frac{\sqrt[18]{ y^{13} }}{|y|}\\---------------\)
11. \(\left(a^{\frac{-5}{4}}\right)^{\frac{-1}{6}}\\= a^{ \frac{-5}{4} . (\frac{-1}{6}) }= a^{\frac{5}{24}}\\=\sqrt[24]{ a^{5} }\\---------------\)
12. \(\left(x^{\frac{-3}{2}}\right)^{\frac{5}{2}}\\= x^{ \frac{-3}{2} . \frac{5}{2} }= x^{\frac{-15}{4}}\\=\frac{1}{\sqrt[4]{ x^{15} }}\\=\frac{1}{|x^{3}|.\sqrt[4]{ x^{3} }}=\frac{1}{|x^{3}|.\sqrt[4]{ x^{3} }} \color{purple}{\frac{\sqrt[4]{ x }}{\sqrt[4]{ x }}} \\=\frac{\sqrt[4]{ x }}{|x^{4}|}\\---------------\)