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Werk uit m.b.v. de rekenregels

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

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Werk uit m.b.v. de rekenregels

Verbetersleutel

1. \(q^{-1}.q^{\frac{1}{2}}\\= q^{ -1 + \frac{1}{2} }= q^{\frac{-1}{2}}\\=\frac{1}{ \sqrt{ q } }=\frac{1}{ \sqrt{ q } }. \color{purple}{\frac{ \sqrt{ q } }{ \sqrt{ q } }} \\=\frac{ \sqrt{ q } }{|q|}\\---------------\)
2. \(q^{\frac{1}{3}}.q^{\frac{-1}{2}}\\= q^{ \frac{1}{3} + (\frac{-1}{2}) }= q^{\frac{-1}{6}}\\=\frac{1}{\sqrt[6]{ q }}=\frac{1}{\sqrt[6]{ q }}. \color{purple}{\frac{\sqrt[6]{ q^{5} }}{\sqrt[6]{ q^{5} }}} \\=\frac{\sqrt[6]{ q^{5} }}{|q|}\\---------------\)
3. \(x^{\frac{-1}{4}}.x^{\frac{1}{3}}\\= x^{ \frac{-1}{4} + \frac{1}{3} }= x^{\frac{1}{12}}\\=\sqrt[12]{ x }\\---------------\)
4. \(y^{-1}.y^{-1}\\= y^{ -1 + (-1) }= y^{-2}\\=\frac{1}{y^{2}}\\---------------\)
5. \(y^{\frac{1}{2}}.y^{\frac{1}{3}}\\= y^{ \frac{1}{2} + \frac{1}{3} }= y^{\frac{5}{6}}\\=\sqrt[6]{ y^{5} }\\---------------\)
6. \(q^{-1}.q^{\frac{-5}{3}}\\= q^{ -1 + (\frac{-5}{3}) }= q^{\frac{-8}{3}}\\=\frac{1}{\sqrt[3]{ q^{8} }}\\=\frac{1}{q^{2}.\sqrt[3]{ q^{2} }}=\frac{1}{q^{2}.\sqrt[3]{ q^{2} }} \color{purple}{\frac{\sqrt[3]{ q }}{\sqrt[3]{ q }}} \\=\frac{\sqrt[3]{ q }}{q^{3}}\\---------------\)
7. \(q^{-1}.q^{\frac{-4}{3}}\\= q^{ -1 + (\frac{-4}{3}) }= q^{\frac{-7}{3}}\\=\frac{1}{\sqrt[3]{ q^{7} }}\\=\frac{1}{q^{2}.\sqrt[3]{ q }}=\frac{1}{q^{2}.\sqrt[3]{ q }} \color{purple}{\frac{\sqrt[3]{ q^{2} }}{\sqrt[3]{ q^{2} }}} \\=\frac{\sqrt[3]{ q^{2} }}{q^{3}}\\---------------\)
8. \(a^{\frac{-5}{3}}.a^{\frac{3}{4}}\\= a^{ \frac{-5}{3} + \frac{3}{4} }= a^{\frac{-11}{12}}\\=\frac{1}{\sqrt[12]{ a^{11} }}=\frac{1}{\sqrt[12]{ a^{11} }}. \color{purple}{\frac{\sqrt[12]{ a }}{\sqrt[12]{ a }}} \\=\frac{\sqrt[12]{ a }}{|a|}\\---------------\)
9. \(a^{\frac{-2}{5}}.a^{\frac{-1}{3}}\\= a^{ \frac{-2}{5} + (\frac{-1}{3}) }= a^{\frac{-11}{15}}\\=\frac{1}{\sqrt[15]{ a^{11} }}=\frac{1}{\sqrt[15]{ a^{11} }}. \color{purple}{\frac{\sqrt[15]{ a^{4} }}{\sqrt[15]{ a^{4} }}} \\=\frac{\sqrt[15]{ a^{4} }}{a}\\---------------\)
10. \(q^{\frac{2}{5}}.q^{\frac{-1}{2}}\\= q^{ \frac{2}{5} + (\frac{-1}{2}) }= q^{\frac{-1}{10}}\\=\frac{1}{\sqrt[10]{ q }}=\frac{1}{\sqrt[10]{ q }}. \color{purple}{\frac{\sqrt[10]{ q^{9} }}{\sqrt[10]{ q^{9} }}} \\=\frac{\sqrt[10]{ q^{9} }}{|q|}\\---------------\)
11. \(y^{\frac{-5}{4}}.y^{\frac{-1}{4}}\\= y^{ \frac{-5}{4} + (\frac{-1}{4}) }= y^{\frac{-3}{2}}\\=\frac{1}{ \sqrt{ y^{3} } }\\=\frac{1}{|y|. \sqrt{ y } }=\frac{1}{|y|. \sqrt{ y } } \color{purple}{\frac{ \sqrt{ y } }{ \sqrt{ y } }} \\=\frac{ \sqrt{ y } }{|y^{2}|}\\---------------\)
12. \(y^{\frac{5}{3}}.y^{\frac{4}{5}}\\= y^{ \frac{5}{3} + \frac{4}{5} }= y^{\frac{37}{15}}\\=\sqrt[15]{ y^{37} }=y^{2}.\sqrt[15]{ y^{7} }\\---------------\)