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Werk uit m.b.v. de rekenregels

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

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Werk uit m.b.v. de rekenregels

Verbetersleutel

1. \(\left(x^{\frac{5}{4}}\right)^{\frac{-1}{2}}\\= x^{ \frac{5}{4} . (\frac{-1}{2}) }= x^{\frac{-5}{8}}\\=\frac{1}{\sqrt[8]{ x^{5} }}=\frac{1}{\sqrt[8]{ x^{5} }}. \color{purple}{\frac{\sqrt[8]{ x^{3} }}{\sqrt[8]{ x^{3} }}} \\=\frac{\sqrt[8]{ x^{3} }}{|x|}\\---------------\)
2. \(\left(y^{\frac{-1}{4}}\right)^{\frac{3}{4}}\\= y^{ \frac{-1}{4} . \frac{3}{4} }= y^{\frac{-3}{16}}\\=\frac{1}{\sqrt[16]{ y^{3} }}=\frac{1}{\sqrt[16]{ y^{3} }}. \color{purple}{\frac{\sqrt[16]{ y^{13} }}{\sqrt[16]{ y^{13} }}} \\=\frac{\sqrt[16]{ y^{13} }}{|y|}\\---------------\)
3. \(\left(a^{\frac{1}{2}}\right)^{\frac{1}{2}}\\= a^{ \frac{1}{2} . \frac{1}{2} }= a^{\frac{1}{4}}\\=\sqrt[4]{ a }\\---------------\)
4. \(\left(x^{\frac{-1}{2}}\right)^{\frac{4}{3}}\\= x^{ \frac{-1}{2} . \frac{4}{3} }= x^{\frac{-2}{3}}\\=\frac{1}{\sqrt[3]{ x^{2} }}=\frac{1}{\sqrt[3]{ x^{2} }}. \color{purple}{\frac{\sqrt[3]{ x }}{\sqrt[3]{ x }}} \\=\frac{\sqrt[3]{ x }}{x}\\---------------\)
5. \(\left(a^{\frac{1}{2}}\right)^{\frac{1}{3}}\\= a^{ \frac{1}{2} . \frac{1}{3} }= a^{\frac{1}{6}}\\=\sqrt[6]{ a }\\---------------\)
6. \(\left(a^{\frac{-3}{4}}\right)^{\frac{1}{4}}\\= a^{ \frac{-3}{4} . \frac{1}{4} }= a^{\frac{-3}{16}}\\=\frac{1}{\sqrt[16]{ a^{3} }}=\frac{1}{\sqrt[16]{ a^{3} }}. \color{purple}{\frac{\sqrt[16]{ a^{13} }}{\sqrt[16]{ a^{13} }}} \\=\frac{\sqrt[16]{ a^{13} }}{|a|}\\---------------\)
7. \(\left(a^{-1}\right)^{\frac{-5}{2}}\\= a^{ -1 . (\frac{-5}{2}) }= a^{\frac{5}{2}}\\= \sqrt{ a^{5} } =|a^{2}|. \sqrt{ a } \\---------------\)
8. \(\left(q^{\frac{4}{5}}\right)^{\frac{-4}{3}}\\= q^{ \frac{4}{5} . (\frac{-4}{3}) }= q^{\frac{-16}{15}}\\=\frac{1}{\sqrt[15]{ q^{16} }}\\=\frac{1}{q.\sqrt[15]{ q }}=\frac{1}{q.\sqrt[15]{ q }} \color{purple}{\frac{\sqrt[15]{ q^{14} }}{\sqrt[15]{ q^{14} }}} \\=\frac{\sqrt[15]{ q^{14} }}{q^{2}}\\---------------\)
9. \(\left(y^{\frac{-4}{3}}\right)^{1}\\= y^{ \frac{-4}{3} . 1 }= y^{\frac{-4}{3}}\\=\frac{1}{\sqrt[3]{ y^{4} }}\\=\frac{1}{y.\sqrt[3]{ y }}=\frac{1}{y.\sqrt[3]{ y }} \color{purple}{\frac{\sqrt[3]{ y^{2} }}{\sqrt[3]{ y^{2} }}} \\=\frac{\sqrt[3]{ y^{2} }}{y^{2}}\\---------------\)
10. \(\left(q^{\frac{1}{2}}\right)^{\frac{-3}{4}}\\= q^{ \frac{1}{2} . (\frac{-3}{4}) }= q^{\frac{-3}{8}}\\=\frac{1}{\sqrt[8]{ q^{3} }}=\frac{1}{\sqrt[8]{ q^{3} }}. \color{purple}{\frac{\sqrt[8]{ q^{5} }}{\sqrt[8]{ q^{5} }}} \\=\frac{\sqrt[8]{ q^{5} }}{|q|}\\---------------\)
11. \(\left(q^{\frac{-5}{6}}\right)^{\frac{-1}{6}}\\= q^{ \frac{-5}{6} . (\frac{-1}{6}) }= q^{\frac{5}{36}}\\=\sqrt[36]{ q^{5} }\\---------------\)
12. \(\left(y^{2}\right)^{\frac{-4}{3}}\\= y^{ 2 . (\frac{-4}{3}) }= y^{\frac{-8}{3}}\\=\frac{1}{\sqrt[3]{ y^{8} }}\\=\frac{1}{y^{2}.\sqrt[3]{ y^{2} }}=\frac{1}{y^{2}.\sqrt[3]{ y^{2} }} \color{purple}{\frac{\sqrt[3]{ y }}{\sqrt[3]{ y }}} \\=\frac{\sqrt[3]{ y }}{y^{3}}\\---------------\)