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Werk uit m.b.v. de rekenregels

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

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Werk uit m.b.v. de rekenregels

Verbetersleutel

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