Quotiënt zelfde grondtal

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Werk uit m.b.v. de rekenregels

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

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Werk uit m.b.v. de rekenregels

Verbetersleutel

1. \(\dfrac{x^{\frac{1}{2}}}{x^{\frac{-1}{3}}}\\= x^{ \frac{1}{2} - (\frac{-1}{3}) }= x^{\frac{5}{6}}\\=\sqrt[6]{ x^{5} }\\---------------\)
2. \(\dfrac{x^{\frac{-2}{3}}}{x^{\frac{1}{3}}}\\= x^{ \frac{-2}{3} - \frac{1}{3} }= x^{-1}\\=\frac{1}{x}\\---------------\)
3. \(\dfrac{y^{\frac{-2}{3}}}{y^{\frac{-1}{6}}}\\= y^{ \frac{-2}{3} - (\frac{-1}{6}) }= y^{\frac{-1}{2}}\\=\frac{1}{ \sqrt{ y } }=\frac{1}{ \sqrt{ y } }. \color{purple}{\frac{ \sqrt{ y } }{ \sqrt{ y } }} \\=\frac{ \sqrt{ y } }{|y|}\\---------------\)
4. \(\dfrac{q^{1}}{q^{\frac{5}{2}}}\\= q^{ 1 - \frac{5}{2} }= q^{\frac{-3}{2}}\\=\frac{1}{ \sqrt{ q^{3} } }\\=\frac{1}{|q|. \sqrt{ q } }=\frac{1}{|q|. \sqrt{ q } } \color{purple}{\frac{ \sqrt{ q } }{ \sqrt{ q } }} \\=\frac{ \sqrt{ q } }{|q^{2}|}\\---------------\)
5. \(\dfrac{a^{\frac{2}{5}}}{a^{\frac{1}{2}}}\\= a^{ \frac{2}{5} - \frac{1}{2} }= a^{\frac{-1}{10}}\\=\frac{1}{\sqrt[10]{ a }}=\frac{1}{\sqrt[10]{ a }}. \color{purple}{\frac{\sqrt[10]{ a^{9} }}{\sqrt[10]{ a^{9} }}} \\=\frac{\sqrt[10]{ a^{9} }}{|a|}\\---------------\)
6. \(\dfrac{x^{\frac{2}{3}}}{x^{\frac{-5}{3}}}\\= x^{ \frac{2}{3} - (\frac{-5}{3}) }= x^{\frac{7}{3}}\\=\sqrt[3]{ x^{7} }=x^{2}.\sqrt[3]{ x }\\---------------\)
7. \(\dfrac{q^{\frac{1}{5}}}{q^{\frac{4}{3}}}\\= q^{ \frac{1}{5} - \frac{4}{3} }= q^{\frac{-17}{15}}\\=\frac{1}{\sqrt[15]{ q^{17} }}\\=\frac{1}{q.\sqrt[15]{ q^{2} }}=\frac{1}{q.\sqrt[15]{ q^{2} }} \color{purple}{\frac{\sqrt[15]{ q^{13} }}{\sqrt[15]{ q^{13} }}} \\=\frac{\sqrt[15]{ q^{13} }}{q^{2}}\\---------------\)
8. \(\dfrac{q^{-1}}{q^{\frac{-5}{2}}}\\= q^{ -1 - (\frac{-5}{2}) }= q^{\frac{3}{2}}\\= \sqrt{ q^{3} } =|q|. \sqrt{ q } \\---------------\)
9. \(\dfrac{q^{\frac{-3}{4}}}{q^{\frac{-2}{3}}}\\= q^{ \frac{-3}{4} - (\frac{-2}{3}) }= q^{\frac{-1}{12}}\\=\frac{1}{\sqrt[12]{ q }}=\frac{1}{\sqrt[12]{ q }}. \color{purple}{\frac{\sqrt[12]{ q^{11} }}{\sqrt[12]{ q^{11} }}} \\=\frac{\sqrt[12]{ q^{11} }}{|q|}\\---------------\)
10. \(\dfrac{a^{\frac{2}{5}}}{a^{\frac{5}{2}}}\\= a^{ \frac{2}{5} - \frac{5}{2} }= a^{\frac{-21}{10}}\\=\frac{1}{\sqrt[10]{ a^{21} }}\\=\frac{1}{|a^{2}|.\sqrt[10]{ a }}=\frac{1}{|a^{2}|.\sqrt[10]{ a }} \color{purple}{\frac{\sqrt[10]{ a^{9} }}{\sqrt[10]{ a^{9} }}} \\=\frac{\sqrt[10]{ a^{9} }}{|a^{3}|}\\---------------\)
11. \(\dfrac{a^{\frac{-1}{6}}}{a^{\frac{-1}{6}}}\\= a^{ \frac{-1}{6} - (\frac{-1}{6}) }= a^{0}\\=1\\---------------\)
12. \(\dfrac{x^{\frac{-1}{3}}}{x^{-1}}\\= x^{ \frac{-1}{3} - (-1) }= x^{\frac{2}{3}}\\=\sqrt[3]{ x^{2} }\\---------------\)