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Werk uit m.b.v. de rekenregels

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

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Werk uit m.b.v. de rekenregels

Verbetersleutel

1. \(\left(x^{\frac{-2}{3}}\right)^{\frac{-2}{3}}\\= x^{ \frac{-2}{3} . (\frac{-2}{3}) }= x^{\frac{4}{9}}\\=\sqrt[9]{ x^{4} }\\---------------\)
2. \(\left(q^{\frac{3}{5}}\right)^{-1}\\= q^{ \frac{3}{5} . (-1) }= q^{\frac{-3}{5}}\\=\frac{1}{\sqrt[5]{ q^{3} }}=\frac{1}{\sqrt[5]{ q^{3} }}. \color{purple}{\frac{\sqrt[5]{ q^{2} }}{\sqrt[5]{ q^{2} }}} \\=\frac{\sqrt[5]{ q^{2} }}{q}\\---------------\)
3. \(\left(q^{\frac{-5}{4}}\right)^{\frac{-1}{6}}\\= q^{ \frac{-5}{4} . (\frac{-1}{6}) }= q^{\frac{5}{24}}\\=\sqrt[24]{ q^{5} }\\---------------\)
4. \(\left(x^{\frac{3}{2}}\right)^{\frac{4}{3}}\\= x^{ \frac{3}{2} . \frac{4}{3} }= x^{2}\\\\---------------\)
5. \(\left(y^{\frac{1}{2}}\right)^{\frac{3}{4}}\\= y^{ \frac{1}{2} . \frac{3}{4} }= y^{\frac{3}{8}}\\=\sqrt[8]{ y^{3} }\\---------------\)
6. \(\left(x^{\frac{-1}{3}}\right)^{1}\\= x^{ \frac{-1}{3} . 1 }= x^{\frac{-1}{3}}\\=\frac{1}{\sqrt[3]{ x }}=\frac{1}{\sqrt[3]{ x }}. \color{purple}{\frac{\sqrt[3]{ x^{2} }}{\sqrt[3]{ x^{2} }}} \\=\frac{\sqrt[3]{ x^{2} }}{x}\\---------------\)
7. \(\left(a^{-1}\right)^{\frac{1}{3}}\\= a^{ -1 . \frac{1}{3} }= a^{\frac{-1}{3}}\\=\frac{1}{\sqrt[3]{ a }}=\frac{1}{\sqrt[3]{ a }}. \color{purple}{\frac{\sqrt[3]{ a^{2} }}{\sqrt[3]{ a^{2} }}} \\=\frac{\sqrt[3]{ a^{2} }}{a}\\---------------\)
8. \(\left(q^{\frac{-1}{4}}\right)^{\frac{1}{6}}\\= q^{ \frac{-1}{4} . \frac{1}{6} }= q^{\frac{-1}{24}}\\=\frac{1}{\sqrt[24]{ q }}=\frac{1}{\sqrt[24]{ q }}. \color{purple}{\frac{\sqrt[24]{ q^{23} }}{\sqrt[24]{ q^{23} }}} \\=\frac{\sqrt[24]{ q^{23} }}{|q|}\\---------------\)
9. \(\left(q^{1}\right)^{-1}\\= q^{ 1 . (-1) }= q^{-1}\\=\frac{1}{q}\\---------------\)
10. \(\left(q^{\frac{4}{3}}\right)^{\frac{4}{3}}\\= q^{ \frac{4}{3} . \frac{4}{3} }= q^{\frac{16}{9}}\\=\sqrt[9]{ q^{16} }=q.\sqrt[9]{ q^{7} }\\---------------\)
11. \(\left(x^{\frac{3}{4}}\right)^{-1}\\= x^{ \frac{3}{4} . (-1) }= x^{\frac{-3}{4}}\\=\frac{1}{\sqrt[4]{ x^{3} }}=\frac{1}{\sqrt[4]{ x^{3} }}. \color{purple}{\frac{\sqrt[4]{ x }}{\sqrt[4]{ x }}} \\=\frac{\sqrt[4]{ x }}{|x|}\\---------------\)
12. \(\left(x^{-1}\right)^{\frac{-4}{3}}\\= x^{ -1 . (\frac{-4}{3}) }= x^{\frac{4}{3}}\\=\sqrt[3]{ x^{4} }=x.\sqrt[3]{ x }\\---------------\)