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Werk uit m.b.v. de rekenregels

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

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Werk uit m.b.v. de rekenregels

Verbetersleutel

1. \(\left(y^{\frac{-1}{2}}\right)^{\frac{1}{2}}\\= y^{ \frac{-1}{2} . \frac{1}{2} }= y^{\frac{-1}{4}}\\=\frac{1}{\sqrt[4]{ y }}=\frac{1}{\sqrt[4]{ y }}. \color{purple}{\frac{\sqrt[4]{ y^{3} }}{\sqrt[4]{ y^{3} }}} \\=\frac{\sqrt[4]{ y^{3} }}{|y|}\\---------------\)
2. \(\left(y^{\frac{-1}{2}}\right)^{1}\\= y^{ \frac{-1}{2} . 1 }= y^{\frac{-1}{2}}\\=\frac{1}{ \sqrt{ y } }=\frac{1}{ \sqrt{ y } }. \color{purple}{\frac{ \sqrt{ y } }{ \sqrt{ y } }} \\=\frac{ \sqrt{ y } }{|y|}\\---------------\)
3. \(\left(a^{\frac{1}{3}}\right)^{1}\\= a^{ \frac{1}{3} . 1 }= a^{\frac{1}{3}}\\=\sqrt[3]{ a }\\---------------\)
4. \(\left(x^{\frac{5}{3}}\right)^{\frac{4}{3}}\\= x^{ \frac{5}{3} . \frac{4}{3} }= x^{\frac{20}{9}}\\=\sqrt[9]{ x^{20} }=x^{2}.\sqrt[9]{ x^{2} }\\---------------\)
5. \(\left(a^{\frac{1}{5}}\right)^{-1}\\= a^{ \frac{1}{5} . (-1) }= a^{\frac{-1}{5}}\\=\frac{1}{\sqrt[5]{ a }}=\frac{1}{\sqrt[5]{ a }}. \color{purple}{\frac{\sqrt[5]{ a^{4} }}{\sqrt[5]{ a^{4} }}} \\=\frac{\sqrt[5]{ a^{4} }}{a}\\---------------\)
6. \(\left(x^{\frac{2}{5}}\right)^{\frac{-3}{5}}\\= x^{ \frac{2}{5} . (\frac{-3}{5}) }= x^{\frac{-6}{25}}\\=\frac{1}{\sqrt[25]{ x^{6} }}=\frac{1}{\sqrt[25]{ x^{6} }}. \color{purple}{\frac{\sqrt[25]{ x^{19} }}{\sqrt[25]{ x^{19} }}} \\=\frac{\sqrt[25]{ x^{19} }}{x}\\---------------\)
7. \(\left(y^{\frac{4}{5}}\right)^{\frac{5}{3}}\\= y^{ \frac{4}{5} . \frac{5}{3} }= y^{\frac{4}{3}}\\=\sqrt[3]{ y^{4} }=y.\sqrt[3]{ y }\\---------------\)
8. \(\left(y^{\frac{1}{2}}\right)^{\frac{-2}{5}}\\= y^{ \frac{1}{2} . (\frac{-2}{5}) }= y^{\frac{-1}{5}}\\=\frac{1}{\sqrt[5]{ y }}=\frac{1}{\sqrt[5]{ y }}. \color{purple}{\frac{\sqrt[5]{ y^{4} }}{\sqrt[5]{ y^{4} }}} \\=\frac{\sqrt[5]{ y^{4} }}{y}\\---------------\)
9. \(\left(a^{\frac{5}{2}}\right)^{\frac{-5}{2}}\\= a^{ \frac{5}{2} . (\frac{-5}{2}) }= a^{\frac{-25}{4}}\\=\frac{1}{\sqrt[4]{ a^{25} }}\\=\frac{1}{|a^{6}|.\sqrt[4]{ a }}=\frac{1}{|a^{6}|.\sqrt[4]{ a }} \color{purple}{\frac{\sqrt[4]{ a^{3} }}{\sqrt[4]{ a^{3} }}} \\=\frac{\sqrt[4]{ a^{3} }}{|a^{7}|}\\---------------\)
10. \(\left(x^{\frac{1}{3}}\right)^{\frac{-5}{4}}\\= x^{ \frac{1}{3} . (\frac{-5}{4}) }= x^{\frac{-5}{12}}\\=\frac{1}{\sqrt[12]{ x^{5} }}=\frac{1}{\sqrt[12]{ x^{5} }}. \color{purple}{\frac{\sqrt[12]{ x^{7} }}{\sqrt[12]{ x^{7} }}} \\=\frac{\sqrt[12]{ x^{7} }}{|x|}\\---------------\)
11. \(\left(x^{\frac{1}{3}}\right)^{\frac{1}{3}}\\= x^{ \frac{1}{3} . \frac{1}{3} }= x^{\frac{1}{9}}\\=\sqrt[9]{ x }\\---------------\)
12. \(\left(x^{\frac{-5}{2}}\right)^{\frac{3}{4}}\\= x^{ \frac{-5}{2} . \frac{3}{4} }= x^{\frac{-15}{8}}\\=\frac{1}{\sqrt[8]{ x^{15} }}\\=\frac{1}{|x|.\sqrt[8]{ x^{7} }}=\frac{1}{|x|.\sqrt[8]{ x^{7} }} \color{purple}{\frac{\sqrt[8]{ x }}{\sqrt[8]{ x }}} \\=\frac{\sqrt[8]{ x }}{|x^{2}|}\\---------------\)