Quotiënt zelfde grondtal

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Werk uit m.b.v. de rekenregels

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

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Werk uit m.b.v. de rekenregels

Verbetersleutel

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