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Werk uit m.b.v. de rekenregels

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

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Werk uit m.b.v. de rekenregels

Verbetersleutel

1. \(\left(y^{\frac{1}{3}}\right)^{-1}\\= y^{ \frac{1}{3} . (-1) }= y^{\frac{-1}{3}}\\=\frac{1}{\sqrt[3]{ y }}=\frac{1}{\sqrt[3]{ y }}. \color{purple}{\frac{\sqrt[3]{ y^{2} }}{\sqrt[3]{ y^{2} }}} \\=\frac{\sqrt[3]{ y^{2} }}{y}\\---------------\)
2. \(\left(x^{\frac{1}{5}}\right)^{\frac{-3}{5}}\\= x^{ \frac{1}{5} . (\frac{-3}{5}) }= x^{\frac{-3}{25}}\\=\frac{1}{\sqrt[25]{ x^{3} }}=\frac{1}{\sqrt[25]{ x^{3} }}. \color{purple}{\frac{\sqrt[25]{ x^{22} }}{\sqrt[25]{ x^{22} }}} \\=\frac{\sqrt[25]{ x^{22} }}{x}\\---------------\)
3. \(\left(q^{\frac{4}{5}}\right)^{\frac{1}{4}}\\= q^{ \frac{4}{5} . \frac{1}{4} }= q^{\frac{1}{5}}\\=\sqrt[5]{ q }\\---------------\)
4. \(\left(y^{\frac{-4}{3}}\right)^{2}\\= y^{ \frac{-4}{3} . 2 }= y^{\frac{-8}{3}}\\=\frac{1}{\sqrt[3]{ y^{8} }}\\=\frac{1}{y^{2}.\sqrt[3]{ y^{2} }}=\frac{1}{y^{2}.\sqrt[3]{ y^{2} }} \color{purple}{\frac{\sqrt[3]{ y }}{\sqrt[3]{ y }}} \\=\frac{\sqrt[3]{ y }}{y^{3}}\\---------------\)
5. \(\left(x^{\frac{-1}{2}}\right)^{\frac{3}{2}}\\= x^{ \frac{-1}{2} . \frac{3}{2} }= x^{\frac{-3}{4}}\\=\frac{1}{\sqrt[4]{ x^{3} }}=\frac{1}{\sqrt[4]{ x^{3} }}. \color{purple}{\frac{\sqrt[4]{ x }}{\sqrt[4]{ x }}} \\=\frac{\sqrt[4]{ x }}{|x|}\\---------------\)
6. \(\left(y^{\frac{-1}{2}}\right)^{1}\\= y^{ \frac{-1}{2} . 1 }= y^{\frac{-1}{2}}\\=\frac{1}{ \sqrt{ y } }=\frac{1}{ \sqrt{ y } }. \color{purple}{\frac{ \sqrt{ y } }{ \sqrt{ y } }} \\=\frac{ \sqrt{ y } }{|y|}\\---------------\)
7. \(\left(x^{\frac{-2}{3}}\right)^{\frac{1}{2}}\\= x^{ \frac{-2}{3} . \frac{1}{2} }= x^{\frac{-1}{3}}\\=\frac{1}{\sqrt[3]{ x }}=\frac{1}{\sqrt[3]{ x }}. \color{purple}{\frac{\sqrt[3]{ x^{2} }}{\sqrt[3]{ x^{2} }}} \\=\frac{\sqrt[3]{ x^{2} }}{x}\\---------------\)
8. \(\left(a^{\frac{-1}{2}}\right)^{\frac{4}{3}}\\= a^{ \frac{-1}{2} . \frac{4}{3} }= a^{\frac{-2}{3}}\\=\frac{1}{\sqrt[3]{ a^{2} }}=\frac{1}{\sqrt[3]{ a^{2} }}. \color{purple}{\frac{\sqrt[3]{ a }}{\sqrt[3]{ a }}} \\=\frac{\sqrt[3]{ a }}{a}\\---------------\)
9. \(\left(a^{-2}\right)^{\frac{-4}{3}}\\= a^{ -2 . (\frac{-4}{3}) }= a^{\frac{8}{3}}\\=\sqrt[3]{ a^{8} }=a^{2}.\sqrt[3]{ a^{2} }\\---------------\)
10. \(\left(q^{-1}\right)^{\frac{1}{2}}\\= q^{ -1 . \frac{1}{2} }= q^{\frac{-1}{2}}\\=\frac{1}{ \sqrt{ q } }=\frac{1}{ \sqrt{ q } }. \color{purple}{\frac{ \sqrt{ q } }{ \sqrt{ q } }} \\=\frac{ \sqrt{ q } }{|q|}\\---------------\)
11. \(\left(y^{\frac{2}{3}}\right)^{2}\\= y^{ \frac{2}{3} . 2 }= y^{\frac{4}{3}}\\=\sqrt[3]{ y^{4} }=y.\sqrt[3]{ y }\\---------------\)
12. \(\left(y^{\frac{-5}{3}}\right)^{-1}\\= y^{ \frac{-5}{3} . (-1) }= y^{\frac{5}{3}}\\=\sqrt[3]{ y^{5} }=y.\sqrt[3]{ y^{2} }\\---------------\)