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Werk uit m.b.v. de rekenregels

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

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Werk uit m.b.v. de rekenregels

Verbetersleutel

1. \(\left(x^{\frac{-5}{3}}\right)^{\frac{-3}{2}}\\= x^{ \frac{-5}{3} . (\frac{-3}{2}) }= x^{\frac{5}{2}}\\= \sqrt{ x^{5} } =|x^{2}|. \sqrt{ x } \\---------------\)
2. \(\left(x^{\frac{-5}{2}}\right)^{\frac{1}{2}}\\= x^{ \frac{-5}{2} . \frac{1}{2} }= x^{\frac{-5}{4}}\\=\frac{1}{\sqrt[4]{ x^{5} }}\\=\frac{1}{|x|.\sqrt[4]{ x }}=\frac{1}{|x|.\sqrt[4]{ x }} \color{purple}{\frac{\sqrt[4]{ x^{3} }}{\sqrt[4]{ x^{3} }}} \\=\frac{\sqrt[4]{ x^{3} }}{|x^{2}|}\\---------------\)
3. \(\left(q^{\frac{1}{6}}\right)^{\frac{-1}{2}}\\= q^{ \frac{1}{6} . (\frac{-1}{2}) }= q^{\frac{-1}{12}}\\=\frac{1}{\sqrt[12]{ q }}=\frac{1}{\sqrt[12]{ q }}. \color{purple}{\frac{\sqrt[12]{ q^{11} }}{\sqrt[12]{ q^{11} }}} \\=\frac{\sqrt[12]{ q^{11} }}{|q|}\\---------------\)
4. \(\left(x^{\frac{1}{3}}\right)^{\frac{-1}{5}}\\= x^{ \frac{1}{3} . (\frac{-1}{5}) }= x^{\frac{-1}{15}}\\=\frac{1}{\sqrt[15]{ x }}=\frac{1}{\sqrt[15]{ x }}. \color{purple}{\frac{\sqrt[15]{ x^{14} }}{\sqrt[15]{ x^{14} }}} \\=\frac{\sqrt[15]{ x^{14} }}{x}\\---------------\)
5. \(\left(q^{1}\right)^{\frac{1}{4}}\\= q^{ 1 . \frac{1}{4} }= q^{\frac{1}{4}}\\=\sqrt[4]{ q }\\---------------\)
6. \(\left(a^{1}\right)^{-1}\\= a^{ 1 . (-1) }= a^{-1}\\=\frac{1}{a}\\---------------\)
7. \(\left(x^{\frac{3}{4}}\right)^{\frac{3}{4}}\\= x^{ \frac{3}{4} . \frac{3}{4} }= x^{\frac{9}{16}}\\=\sqrt[16]{ x^{9} }\\---------------\)
8. \(\left(x^{\frac{1}{3}}\right)^{\frac{-1}{3}}\\= x^{ \frac{1}{3} . (\frac{-1}{3}) }= x^{\frac{-1}{9}}\\=\frac{1}{\sqrt[9]{ x }}=\frac{1}{\sqrt[9]{ x }}. \color{purple}{\frac{\sqrt[9]{ x^{8} }}{\sqrt[9]{ x^{8} }}} \\=\frac{\sqrt[9]{ x^{8} }}{x}\\---------------\)
9. \(\left(y^{\frac{1}{3}}\right)^{\frac{5}{3}}\\= y^{ \frac{1}{3} . \frac{5}{3} }= y^{\frac{5}{9}}\\=\sqrt[9]{ y^{5} }\\---------------\)
10. \(\left(x^{2}\right)^{\frac{-2}{3}}\\= x^{ 2 . (\frac{-2}{3}) }= x^{\frac{-4}{3}}\\=\frac{1}{\sqrt[3]{ x^{4} }}\\=\frac{1}{x.\sqrt[3]{ x }}=\frac{1}{x.\sqrt[3]{ x }} \color{purple}{\frac{\sqrt[3]{ x^{2} }}{\sqrt[3]{ x^{2} }}} \\=\frac{\sqrt[3]{ x^{2} }}{x^{2}}\\---------------\)
11. \(\left(a^{\frac{1}{5}}\right)^{\frac{-5}{3}}\\= a^{ \frac{1}{5} . (\frac{-5}{3}) }= a^{\frac{-1}{3}}\\=\frac{1}{\sqrt[3]{ a }}=\frac{1}{\sqrt[3]{ a }}. \color{purple}{\frac{\sqrt[3]{ a^{2} }}{\sqrt[3]{ a^{2} }}} \\=\frac{\sqrt[3]{ a^{2} }}{a}\\---------------\)
12. \(\left(a^{\frac{1}{2}}\right)^{\frac{1}{4}}\\= a^{ \frac{1}{2} . \frac{1}{4} }= a^{\frac{1}{8}}\\=\sqrt[8]{ a }\\---------------\)