Przedstaw w postaci...- Zadanie 3: Matematyka 7. Ćwiczenia podstawowe

Rozwiązanie

a) $5^{2} \cdot 25 = 5^{2} \cdot 5^{2} = 5^{2 + 2} = 5^{4}$

b) $125 \cdot 5^{6} = 5^{3} \cdot 5^{6} = 5^{3 + 6} = 5^{9}$

c) $\frac{125 ^{8}}{5} = \frac{( 5 ^{3} ) ^{8}}{5 ^{1}} = \frac{5 ^{3 \cdot 8}}{5 ^{1}} = \frac{5 ^{24}}{5 ^{1}} = 5^{24 - 1} = 5^{23}$

d) $5^{18} : 25^{6} = \frac{5 ^{18}}{25 ^{6}} = \frac{5 ^{18}}{( 5 ^{2} ) ^{6}} = \frac{5 ^{18}}{5 ^{2 \cdot 6}} = \frac{5 ^{18}}{5 ^{12}} = 5^{18 - 12} = 5^{6}$

e) $(5^{7} \cdot 25)^{2} = (5^{7} \cdot 5^{2})^{2} = (5^{7 + 2})^{2} = (5^{9})^{2} = 5^{9 \cdot 2} = 5^{18}$

f) $25^{10} : 125^{5} = \frac{25 ^{10}}{125 ^{5}} = \frac{( 5 ^{2} ) ^{10}}{( 5 ^{3} ) ^{5}} = \frac{5 ^{2 \cdot 10}}{5 ^{3 \cdot 5}} = \frac{5 ^{20}}{5 ^{15}} = 5^{20 - 15} = 5^{5}$

g) $(125^{2})^{5} \cdot 5 = ((5^{3})^{2})^{5} \cdot 5^{1} = (5^{3 \cdot 2})^{5} \cdot 5^{1} = (5^{6})^{5} \cdot 5^{1} = 5^{6 \cdot 5} \cdot 5^{1} = 5^{30} \cdot 5^{1} = 5^{30 + 1} = 5^{31}$