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Werk uit m.b.v. de rekenregels

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

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Werk uit m.b.v. de rekenregels

Verbetersleutel

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