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Werk uit m.b.v. de rekenregels

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

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Werk uit m.b.v. de rekenregels

Verbetersleutel

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