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## Cvičení - Mocniny s racionálním mocnitelem

### Cvičení 2.28

Rozhodni, zda platí:
 a) $\displaystyle \sqrt[\large 5 \,]{7^6 \;} = 7^{\Large \frac {6} {5}}$ $\; \; \;$ ano $\,$ ne $\; \; \; \;$ b) $\displaystyle 19^{\Large \frac {12} {7}} = \sqrt[\large 7 \,]{19^{12}\;}$ $\; \; \;$ ano $\,$ ne $\; \; \; \;$ c) $\displaystyle \sqrt{25^3\;} = 25^{\Large \frac {3} {1}}$ $\; \; \;$ ano $\,$ ne $\; \; \; \;$ d) $\displaystyle (0,4)^{\Large \frac {1} {2}} = \sqrt{0,4\;} \; \; \; \; \;$ $\; \; \;$ ano $\,$ ne $\; \; \; \;$ e) $\displaystyle \sqrt[\large 6 \,]{5^{-\,4}\;} = -\,5^{\Large \frac {4} {6}}$ $\; \; \;$ ano $\,$ ne $\; \; \; \;$ f) $\displaystyle \left(\frac {1} {\,15\,}\right)^{\Large -\,\frac {3} {4}} = \frac {\; 1^{-\,3}\,} {\,\sqrt[\large 4 \,]{15\;}\,}$ $\; \; \;$ ano $\,$ ne $\; \; \; \;$

### Cvičení 2.29

Přiřaď:
 $\displaystyle 0,04$ $\displaystyle 0,4$ $\displaystyle 2$ $\displaystyle 3$ $\displaystyle 4$ $\displaystyle 9$ a) $\displaystyle 16^{\Large \frac {1} {4}} = \;$ b) $\displaystyle 16^{\Large \frac {2} {4}} = \;$ c) $\displaystyle 81^{\Large \frac {5} {10}} = \;$ d) $\displaystyle (0,0016)^{\Large \frac {1} {2}} = \;$ e) $\displaystyle (0,064)^{\Large \frac {1} {3}} = \;$ f) $\displaystyle 27^{\Large \frac {2} {6}} = \;$

### Cvičení 2.30

Vypočítej:

a) $\displaystyle 2^{\Large \frac {3} {2}} \cdot 2^{\Large \frac {4} {3}} \cdot 2^{\Large \frac {7} {6}} = \;$

b) $\displaystyle \frac {\left(5^{\Large \frac {3} {5}} \right)^{\Large \frac {3} {2}}} {5^{\Large \frac {4} {10}}} = \;$

c) $\displaystyle \frac {\left(45^{\Large \frac {2} {5}}\right)^{\Large \frac {17} {4}}} {5^{\Large \frac {7} {10}}} \cdot 9^{\Large - \,\frac {6} {5}} = \;$

d) $\displaystyle \frac {\left(3^{\Large \frac {30} {7}} \cdot 4^{\Large \frac {18} {7}} \right)^{\Large \frac {7} {6}}} {36^{\Large \frac {5} {2}}} = \;$

### Cvičení 2.31

Vyjádři ve tvaru jediné odmocniny:

a) $\displaystyle \sqrt{2 \;} \cdot \sqrt[\large 3 \,]{2^4 \;} \cdot \sqrt[\large 6 \,]{2^4 \;} = \;$

b) $\displaystyle \frac {\sqrt[\large 3 \,]{6 \;}} {\sqrt[\large 6 \,]{36 \;}} \cdot \sqrt[\large 4 \,]{6^3 \;} = \;$

c) $\displaystyle \frac {1} {10\,000} \cdot \sqrt[\large 4 \,]{5 \cdot \sqrt[\large 3 \,]{2^4 \cdot 5 \;}\;} = \;$

d) $\displaystyle \frac {\sqrt[\large 5 \,]{144 \;} \cdot \sqrt[\large 10 \,]{4^6 \;}} {\sqrt{27 \;}} \cdot \sqrt[\large 5 \,]{\sqrt{4 \;}\;} = \;$

### Cvičení 2.32

Vypočítej:

a) $\displaystyle \left[5 \cdot \left(5 \cdot 5^4 \right)^{3}\right]^{\Large \frac {3} {4}} = \;$

b) $\displaystyle 2^{\Large \frac {4} {3}} \cdot \left(\frac {1} {2} \right)^3 \cdot \left(2^{\Large \frac {3} {2}} \right)^4 = \;$

c) $\displaystyle \left(\frac {4} {27} \right)^{\Large \frac {1} {3}} \div \left(\frac {32} {81} \right)^{\Large \frac {1} {4}} = \;$

d) $\displaystyle \frac {3^{\Large \frac {5} {4}} \cdot 8^{\Large \frac {12}{9}} \cdot (0,5)^{\Large \frac {2} {3}}} {\left(\large \frac {5} {10} \right)^{\Large \frac {6} {18}} \cdot 3^{\Large \frac {1} {4}}} = \;$