Tehted astmete ja juurtega

Murrulise astendaja korral kehtivad kõik viis astendamise põhivalemit.

Näiteid.

$\sqrt[3]{8^{2}} + \sqrt[4]{{(\frac{1}{81})}^{- 3}}$ = $8^{\frac{2}{3}} + {(\frac{1}{81})}^{- \frac{3}{4}}$ = ${(2^{3})}^{\frac{2}{3}} + {(3^{- 4})}^{- \frac{3}{4}}$ = $2^{3 \cdot \frac{2}{3}} + 3^{- 4 \cdot (- \frac{3}{4})}$ = $2^{2} + 3^{3}$ = $31$
$\sqrt{\sqrt[5]{2^{10} \cdot 3^{5}}}$ = ${[{(2^{10} \cdot 3^{5})}^{\frac{1}{5}}]}^{\frac{1}{2}}$ = ${(2^{10} \cdot 3^{5})}^{\frac{1}{10}}$ = $2^{10 \cdot \frac{1}{10}} \cdot 3^{5 \cdot \frac{1}{10}}$ = $2 \cdot 3^{\frac{1}{2}}$ = $2 \sqrt{3}$

Ülesanded

$2^{13} \cdot {1,25}^{13} \cdot {0,4}^{13}$ =

49^{4} \cdot {(\frac{1}{7})}^{4} \cdot 16^{2}

{(\frac{3}{4})}^{2} + \frac{3^{2}}{4} - \frac{3}{4^{2}}

\frac{2^{3}}{3} - \frac{2}{3^{2}} + {(\frac{2}{3})}^{4}

\frac{5^{2} 5^{- 1} - 5^{0}}{2^{- 2}}

\frac{3^{2} 3^{- 1} + {1,5}^{0}}{2^{- 2}}

\frac{4^{- 1} - 3 \cdot {(\frac{2}{3})}^{- 2}}{5 - {(\frac{1}{2})}^{- 1}}

{[\frac{3}{4} - {(\frac{2}{3})}^{- 1}]}^{- 2}

\sqrt[3]{\sqrt{\sqrt{6}}}

\sqrt[3]{\sqrt[4]{125}}

\sqrt[4]{\sqrt{\sqrt[3]{16}}}

\sqrt[3]{2000} + 4 \cdot \sqrt[3]{\frac{1}{32}} - 2 \cdot \sqrt[3]{- 2} + 3 \cdot \sqrt[3]{2}

\sqrt[6]{16} (\sqrt{8} - 10 \cdot \sqrt[3]{32} + \sqrt[6]{2})

\sqrt{2} (\sqrt{2} + \sqrt[3]{4} + \sqrt[4]{8})

(\sqrt{3} + \sqrt[3]{2}) (\sqrt{2} - \sqrt[3]{3})

(1 - \sqrt[3]{5}) (1 + \sqrt[3]{5} + \sqrt[3]{25})

49^{\frac{1}{2}} - 64^{\frac{1}{3}} + 125^{\frac{2}{3}}

$25^{\frac{1}{2}} - 27^{\frac{2}{3}} + 81^{\frac{3}{4}}$ =

${[\frac{\sqrt[3]{32} \cdot 2^{- \frac{2}{3}}}{{(3 - 2)}^{0}}]}^{- 5}$ =

{(\frac{\sqrt[5]{64} \cdot 2^{- \frac{1}{5}}}{\sqrt{8}})}^{- 4}

{[{(\frac{3}{4})}^{0}]}^{- 0,5} - 7,5 \cdot 4^{- \frac{3}{2}} - {(- 2)}^{- 4} + 81^{0,25}

${[{(\frac{1}{4})}^{- 1}]}^{\frac{1}{2}} + {(\frac{8}{27})}^{- \frac{2}{3}} - 81^{\frac{3}{4}} + {(- 16)}^{0}$ =

{(\frac{1}{3})}^{- 10} \cdot 27^{- 3} + {0,2}^{- 4} \cdot 25^{- 2} + {(64^{- \frac{1}{9}})}^{- 3}