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Werk uit m.b.v. de rekenregels

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

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Werk uit m.b.v. de rekenregels

Verbetersleutel

1. \(\left(a^{\frac{5}{3}}\right)^{1}\\= a^{ \frac{5}{3} . 1 }= a^{\frac{5}{3}}\\=\sqrt[3]{ a^{5} }=a.\sqrt[3]{ a^{2} }\\---------------\)
2. \(\left(y^{\frac{5}{6}}\right)^{\frac{-3}{4}}\\= y^{ \frac{5}{6} . (\frac{-3}{4}) }= y^{\frac{-5}{8}}\\=\frac{1}{\sqrt[8]{ y^{5} }}=\frac{1}{\sqrt[8]{ y^{5} }}. \color{purple}{\frac{\sqrt[8]{ y^{3} }}{\sqrt[8]{ y^{3} }}} \\=\frac{\sqrt[8]{ y^{3} }}{|y|}\\---------------\)
3. \(\left(y^{-1}\right)^{\frac{1}{2}}\\= y^{ -1 . \frac{1}{2} }= y^{\frac{-1}{2}}\\=\frac{1}{ \sqrt{ y } }=\frac{1}{ \sqrt{ y } }. \color{purple}{\frac{ \sqrt{ y } }{ \sqrt{ y } }} \\=\frac{ \sqrt{ y } }{|y|}\\---------------\)
4. \(\left(a^{\frac{-5}{2}}\right)^{\frac{1}{3}}\\= a^{ \frac{-5}{2} . \frac{1}{3} }= a^{\frac{-5}{6}}\\=\frac{1}{\sqrt[6]{ a^{5} }}=\frac{1}{\sqrt[6]{ a^{5} }}. \color{purple}{\frac{\sqrt[6]{ a }}{\sqrt[6]{ a }}} \\=\frac{\sqrt[6]{ a }}{|a|}\\---------------\)
5. \(\left(q^{\frac{4}{3}}\right)^{\frac{1}{2}}\\= q^{ \frac{4}{3} . \frac{1}{2} }= q^{\frac{2}{3}}\\=\sqrt[3]{ q^{2} }\\---------------\)
6. \(\left(a^{\frac{4}{5}}\right)^{\frac{-5}{4}}\\= a^{ \frac{4}{5} . (\frac{-5}{4}) }= a^{-1}\\=\frac{1}{a}\\---------------\)
7. \(\left(q^{\frac{1}{6}}\right)^{\frac{-1}{5}}\\= q^{ \frac{1}{6} . (\frac{-1}{5}) }= q^{\frac{-1}{30}}\\=\frac{1}{\sqrt[30]{ q }}=\frac{1}{\sqrt[30]{ q }}. \color{purple}{\frac{\sqrt[30]{ q^{29} }}{\sqrt[30]{ q^{29} }}} \\=\frac{\sqrt[30]{ q^{29} }}{|q|}\\---------------\)
8. \(\left(q^{\frac{2}{3}}\right)^{\frac{4}{5}}\\= q^{ \frac{2}{3} . \frac{4}{5} }= q^{\frac{8}{15}}\\=\sqrt[15]{ q^{8} }\\---------------\)
9. \(\left(y^{\frac{-1}{2}}\right)^{\frac{2}{3}}\\= y^{ \frac{-1}{2} . \frac{2}{3} }= y^{\frac{-1}{3}}\\=\frac{1}{\sqrt[3]{ y }}=\frac{1}{\sqrt[3]{ y }}. \color{purple}{\frac{\sqrt[3]{ y^{2} }}{\sqrt[3]{ y^{2} }}} \\=\frac{\sqrt[3]{ y^{2} }}{y}\\---------------\)
10. \(\left(x^{\frac{3}{5}}\right)^{\frac{3}{4}}\\= x^{ \frac{3}{5} . \frac{3}{4} }= x^{\frac{9}{20}}\\=\sqrt[20]{ x^{9} }\\---------------\)
11. \(\left(a^{-2}\right)^{\frac{5}{3}}\\= a^{ -2 . \frac{5}{3} }= a^{\frac{-10}{3}}\\=\frac{1}{\sqrt[3]{ a^{10} }}\\=\frac{1}{a^{3}.\sqrt[3]{ a }}=\frac{1}{a^{3}.\sqrt[3]{ a }} \color{purple}{\frac{\sqrt[3]{ a^{2} }}{\sqrt[3]{ a^{2} }}} \\=\frac{\sqrt[3]{ a^{2} }}{a^{4}}\\---------------\)
12. \(\left(x^{\frac{-5}{4}}\right)^{\frac{5}{4}}\\= x^{ \frac{-5}{4} . \frac{5}{4} }= x^{\frac{-25}{16}}\\=\frac{1}{\sqrt[16]{ x^{25} }}\\=\frac{1}{|x|.\sqrt[16]{ x^{9} }}=\frac{1}{|x|.\sqrt[16]{ x^{9} }} \color{purple}{\frac{\sqrt[16]{ x^{7} }}{\sqrt[16]{ x^{7} }}} \\=\frac{\sqrt[16]{ x^{7} }}{|x^{2}|}\\---------------\)