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Werk uit m.b.v. de rekenregels

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

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Werk uit m.b.v. de rekenregels

Verbetersleutel

1. \(y^{\frac{1}{3}}.y^{\frac{-2}{3}}\\= y^{ \frac{1}{3} + (\frac{-2}{3}) }= y^{\frac{-1}{3}}\\=\frac{1}{\sqrt[3]{ y }}=\frac{1}{\sqrt[3]{ y }}. \color{purple}{\frac{\sqrt[3]{ y^{2} }}{\sqrt[3]{ y^{2} }}} \\=\frac{\sqrt[3]{ y^{2} }}{y}\\---------------\)
2. \(a^{\frac{1}{5}}.a^{\frac{-1}{6}}\\= a^{ \frac{1}{5} + (\frac{-1}{6}) }= a^{\frac{1}{30}}\\=\sqrt[30]{ a }\\---------------\)
3. \(q^{1}.q^{\frac{-5}{2}}\\= q^{ 1 + (\frac{-5}{2}) }= q^{\frac{-3}{2}}\\=\frac{1}{ \sqrt{ q^{3} } }\\=\frac{1}{|q|. \sqrt{ q } }=\frac{1}{|q|. \sqrt{ q } } \color{purple}{\frac{ \sqrt{ q } }{ \sqrt{ q } }} \\=\frac{ \sqrt{ q } }{|q^{2}|}\\---------------\)
4. \(q^{\frac{-1}{2}}.q^{1}\\= q^{ \frac{-1}{2} + 1 }= q^{\frac{1}{2}}\\= \sqrt{ q } \\---------------\)
5. \(q^{-1}.q^{\frac{5}{2}}\\= q^{ -1 + \frac{5}{2} }= q^{\frac{3}{2}}\\= \sqrt{ q^{3} } =|q|. \sqrt{ q } \\---------------\)
6. \(q^{-1}.q^{\frac{-2}{3}}\\= q^{ -1 + (\frac{-2}{3}) }= q^{\frac{-5}{3}}\\=\frac{1}{\sqrt[3]{ q^{5} }}\\=\frac{1}{q.\sqrt[3]{ q^{2} }}=\frac{1}{q.\sqrt[3]{ q^{2} }} \color{purple}{\frac{\sqrt[3]{ q }}{\sqrt[3]{ q }}} \\=\frac{\sqrt[3]{ q }}{q^{2}}\\---------------\)
7. \(q^{\frac{1}{6}}.q^{\frac{1}{2}}\\= q^{ \frac{1}{6} + \frac{1}{2} }= q^{\frac{2}{3}}\\=\sqrt[3]{ q^{2} }\\---------------\)
8. \(a^{-2}.a^{\frac{-4}{3}}\\= a^{ -2 + (\frac{-4}{3}) }= a^{\frac{-10}{3}}\\=\frac{1}{\sqrt[3]{ a^{10} }}\\=\frac{1}{a^{3}.\sqrt[3]{ a }}=\frac{1}{a^{3}.\sqrt[3]{ a }} \color{purple}{\frac{\sqrt[3]{ a^{2} }}{\sqrt[3]{ a^{2} }}} \\=\frac{\sqrt[3]{ a^{2} }}{a^{4}}\\---------------\)
9. \(q^{1}.q^{\frac{4}{3}}\\= q^{ 1 + \frac{4}{3} }= q^{\frac{7}{3}}\\=\sqrt[3]{ q^{7} }=q^{2}.\sqrt[3]{ q }\\---------------\)
10. \(a^{\frac{-4}{3}}.a^{\frac{-1}{5}}\\= a^{ \frac{-4}{3} + (\frac{-1}{5}) }= a^{\frac{-23}{15}}\\=\frac{1}{\sqrt[15]{ a^{23} }}\\=\frac{1}{a.\sqrt[15]{ a^{8} }}=\frac{1}{a.\sqrt[15]{ a^{8} }} \color{purple}{\frac{\sqrt[15]{ a^{7} }}{\sqrt[15]{ a^{7} }}} \\=\frac{\sqrt[15]{ a^{7} }}{a^{2}}\\---------------\)
11. \(y^{\frac{3}{5}}.y^{\frac{-5}{4}}\\= y^{ \frac{3}{5} + (\frac{-5}{4}) }= y^{\frac{-13}{20}}\\=\frac{1}{\sqrt[20]{ y^{13} }}=\frac{1}{\sqrt[20]{ y^{13} }}. \color{purple}{\frac{\sqrt[20]{ y^{7} }}{\sqrt[20]{ y^{7} }}} \\=\frac{\sqrt[20]{ y^{7} }}{|y|}\\---------------\)
12. \(q^{-2}.q^{\frac{-3}{5}}\\= q^{ -2 + (\frac{-3}{5}) }= q^{\frac{-13}{5}}\\=\frac{1}{\sqrt[5]{ q^{13} }}\\=\frac{1}{q^{2}.\sqrt[5]{ q^{3} }}=\frac{1}{q^{2}.\sqrt[5]{ q^{3} }} \color{purple}{\frac{\sqrt[5]{ q^{2} }}{\sqrt[5]{ q^{2} }}} \\=\frac{\sqrt[5]{ q^{2} }}{q^{3}}\\---------------\)