Nowa MATeMAtyka 1. Podręcznik. Zakres podstawowy | Strona 36
Ćwiczenie 3. Oblicz.
a)
\[2^2 \cdot 2^6 = 2^{2 + 6} = 2^8 = 256\]
b)
\[2^3 \cdot 2^5 = 2^{3 + 5} = 2^8 = 256\]
c)
\[2^9:2^6 = 2^{9 – 6} = 2^3 = 8\]
d)
\[3^9:3^5 = 2^{9 – 5} = 3^4 = 81\]
e)
\[2^6 \cdot 5^6 = (2 \cdot 5)^6 = 10^6 = 1000000\]
f)
\[4^3 \cdot 5^3 = (4 \cdot 5)^3 = 20^3 = 8000\]
g)
\[6^5:3^5 = (6:3)^5 = 2^5 = 32\]
h)
\[12^4:4^4 = (12:4)^4 = 3^4 = 81\]
i)
\[(2^2)^3 = 2^6 = 64\]
j)
\[(3^3)^2 = 3^6 = 729\]
Zadanie 1. Oblicz.
a)
\[(\frac{1}{6})^2 \cdot 6^3 = 6^{- 2} \cdot 6^3 = 6^{- 2 + 3} = 6\]
b)
\[(\frac{1}{5})^3 \cdot 5^5 = 5^{- 3} \cdot 5^5 = 5^{- 3 + 5} = 5^2 = 25\]
c)
\[(\frac{1}{3})^4 \cdot 3^6 = 3^{- 4} \cdot 3^6 = 3^{- 4 + 6} = 3^2 = 9\]
d)
\[( – \frac{1}{2})^5 \cdot ( – 2)^3 = ( – 2)^{- 5} \cdot ( – 2)^3 = ( – 2)^{- 5 + 3} = = ( – 2)^{- 2} = (\frac{1}{2})^2 = \frac{1}{4}\]
e)
\[( – \frac{1}{2})^8 \cdot ( – 2)^5 = ( – \frac{1}{2})^8 \cdot ( – \frac{1}{2})^{- 5} = ( – \frac{1}{2})^{8 – 5} = ( – \frac{1}{2})^3 = – \frac{1}{8}\]
f)
\[( – \frac{1}{4})^3 \cdot ( – 2)^6 = ( – \frac{1}{2})^6 \cdot ( – 2)^6 = – (\frac{1}{2} \cdot 2)^6 = – 1\]
Zadanie 2. Oblicz.
a)
\[(2^2)^5 = 2^{10} = 1024\]
b)
\[(3^2)^2 = 3^4 = 81\]
c)
\[(( – 4)^3)^1 = – 4^3 = – 64\]
d)
\[(( – 5)^2)^2 = 25^2 = 625\]
e)
\[((\frac{1}{2})^3)^3 = (\frac{1}{2})^9 = \frac{1}{512}\]
f)
\[((\frac{1}{3})^3)^2 = (\frac{1}{3})^6 = \frac{1}{729}\]
g)
\[(( – \frac{2}{3})^2)^3 = ( – \frac{2}{3})^6 = \frac{64}{729}\]
h)
\[(( – \frac{3}{2})^3)^2 = ( – \frac{3}{2})^6 = \frac{729}{64} = 11\frac{25}{64}\]
Zadanie 3. Porównaj liczby x i y.
a)
\[x = ( – 4)^2 = 16\]
\[y = ( – \frac{1}{4})^2 = \frac{1}{16}\]
x > y
b)
\[x = ( – \frac{1}{2})^5 = – \frac{1}{32}\]
\[y = ( – \frac{1}{2})^6 = \frac{1}{64}\]
x < y
c)
\[x = ( – \frac{1}{4})^2 = \frac{1}{16}\]
\[y = ( – \frac{1}{4})^4 = \frac{1}{256}\]
x > y
d)
\[x = ( – \frac{2}{3})^3 = – \frac{8}{27}\]
\[y = ( – \frac{3}{2})^3 = – \frac{27}{8}\]
x > y
Zadanie 4. Uporządkuj liczby
\[( – \frac{1}{2})^5,2^4,( – 2)^5,(\frac{1}{2})^3,( – \frac{1}{2})^3,( – \frac{1}{2})^4\]
w kolejności rosnącej.
\[( – \frac{1}{2})^5 = – \frac{1}{32},2^4 = 16,( – 2)^5 = – 32,(\frac{1}{2})^3 = \frac{1}{8},( – \frac{1}{2})^3 = – \frac{1}{8},( – \frac{1}{2})^4 = \frac{1}{16}\]
\[( – 2)^5 < ( – \frac{1}{2})^3 < ( – \frac{1}{2})^5 < ( – \frac{1}{2})^4z < (\frac{1}{2})^3 < ( – \frac{1}{2})^2 < 2^4\]
Zadanie 5. Oblicz.
a)
\[2^5 \cdot 2 \cdot 2^5 = 2^{5 + 1 + 5} = 2^{11} = 2048\]
b)
\[3^2 \cdot 3 \cdot 3^2 = 2^{2 + 1 + 2} = 3^5 = 243\]
c)
\[(4^2 \cdot 4^3):4^5 = 4^{2 + 3 – 5} = 4^0 = 1\]
d)
\[(5^6:5^4) \cdot 5^2 = 4^{6 – 4 + 2} = 5^4 = 625\]
Zadanie 6. Oblicz.
a)
\[5^3 – (2^2)^3 – 3^0 = 125 – 2^6 – 1 = 125 – 64 – 1 = 60\]
b)
\[(4^2)^2 – 4^3 + 4^2 = 4^4 – 4^3 + 4^2 = 256 – 64 + 16 = 208\]
c)
\[11^2 + 2^3 \cdot \frac{1}{2} = 121 + 8 \cdot \frac{1}{2} = 121 + 4 = 125\]
d)
\[3^3 \cdot 2^3 – 36^3:9^3 = 6^3 – 4^3 = 216 – 64 = 152\]
e)
\[( – 2)^4 + ( – 2)^3 = 16 – 8 = 8\]
f)
\[ – 2^5 + ( – 3)^4 = – 32 + 81 = 49\]
g)
\[ 2^4 \cdot 5^4 + 10^3 – 10^2 = 10^4 + 10^3 – 10^2 = 10000 + 1000 – 100 = 10900\]
h)
\[ (3^5 \cdot 2^5):6^4 = 6^5:6^4 = 6\]
i)
\[(6^6:3^6) \cdot 5^6 = 2^6 \cdot 5^6 = 10^6 = 1000000\]