[tex]a) \ 2^2 \cdot 8^{\frac{2}{3}} = 2^4 = 16\\\\b) \ 3^3 \cdot 27 ^{-\frac{4}{3}} = 3^{-1 } = \frac{1}{3}\\\\c)\ 0,008^{\frac{1}{3}} \cdot \sqrt[3]{125} = \sqrt[3]{1} = 1 \\\\d)\ 0,0256 ^{\frac{3}{4}} \cdot (\sqrt[3]{10})^9 = 64[/tex]
Należy wykonać zadanie 5 - obliczyć podane przykłady.
[tex]a^b \cdot a^c = a ^{b + c} \\\\(a^b)^c = a^{b \cdot c} \\\\\sqrt{a} = a^{\frac{1]{2}} \\\\\sqrt[3}{a} = a^{\frac{1}{3}} \\\\a^{-1} =\frac{1}{a} \\\\a^b \cdot c^b = (a \cdot c)^b \\\\a^{\frac{m}{n}} = \sqrt[n]{a^m} \\\\[/tex]
W takim razie:
5.
[tex]a) \\\\[/tex]
[tex]2^2 \cdot 8^{\frac{2}{3}} = 2^2 \cdot (2^3)^{\frac{2}{3}} = 2^2 \cdot 2^{3 \cdot \frac{2}{3}} = 2^2 \cdot 2^2 = 2^{2 + 2} = 2^4 = 16[/tex]
[tex]b) \\\\3^3 \cdot 27 ^{-\frac{4}{3}} = 3^3 \cdot (3^3)^{-\frac{4}{3}} = 3^3 \cdot 3^{3 \cdot (-\frac{4}{3}} = 3^3 \cdot 3^{-4} = 3^{3 - 4} = 3^{-1 } = \frac{1}{3}\\\\c) \\\\0,008^{\frac{1}{3}} \cdot \sqrt[3]{125} = 0,008^{\frac{1}{3}} \cdot 125^{\frac{1}{3}} = (0,008 \cdot 125)^{\frac{1}{3}} = 1^{ \frac{1}{3}} = \sqrt[3]{1} = 1[/tex]
[tex]d) \\\\0,0256 ^{\frac{3}{4}} \cdot (\sqrt[3]{10})^9 = 0,0256^{\frac{3}{4}} \cdot (10^{\frac{1}{3}})^9= 0,0256^{\frac{3}{4}} \cdot 10^3 =\sqrt[4]{0,0256^3} \cdot 10^3 =\\\\ = 0,064 \cdot 1000 = 64[/tex]
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[tex]a) \ 2^2 \cdot 8^{\frac{2}{3}} = 2^4 = 16\\\\b) \ 3^3 \cdot 27 ^{-\frac{4}{3}} = 3^{-1 } = \frac{1}{3}\\\\c)\ 0,008^{\frac{1}{3}} \cdot \sqrt[3]{125} = \sqrt[3]{1} = 1 \\\\d)\ 0,0256 ^{\frac{3}{4}} \cdot (\sqrt[3]{10})^9 = 64[/tex]
Działania na potęgach i pierwiastkach
Należy wykonać zadanie 5 - obliczyć podane przykłady.
Przydatne wzory - potęgowanie i pierwiastkowanie:
[tex]a^b \cdot a^c = a ^{b + c} \\\\(a^b)^c = a^{b \cdot c} \\\\\sqrt{a} = a^{\frac{1]{2}} \\\\\sqrt[3}{a} = a^{\frac{1}{3}} \\\\a^{-1} =\frac{1}{a} \\\\a^b \cdot c^b = (a \cdot c)^b \\\\a^{\frac{m}{n}} = \sqrt[n]{a^m} \\\\[/tex]
W takim razie:
5.
[tex]a) \\\\[/tex]
[tex]2^2 \cdot 8^{\frac{2}{3}} = 2^2 \cdot (2^3)^{\frac{2}{3}} = 2^2 \cdot 2^{3 \cdot \frac{2}{3}} = 2^2 \cdot 2^2 = 2^{2 + 2} = 2^4 = 16[/tex]
[tex]b) \\\\3^3 \cdot 27 ^{-\frac{4}{3}} = 3^3 \cdot (3^3)^{-\frac{4}{3}} = 3^3 \cdot 3^{3 \cdot (-\frac{4}{3}} = 3^3 \cdot 3^{-4} = 3^{3 - 4} = 3^{-1 } = \frac{1}{3}\\\\c) \\\\0,008^{\frac{1}{3}} \cdot \sqrt[3]{125} = 0,008^{\frac{1}{3}} \cdot 125^{\frac{1}{3}} = (0,008 \cdot 125)^{\frac{1}{3}} = 1^{ \frac{1}{3}} = \sqrt[3]{1} = 1[/tex]
[tex]d) \\\\0,0256 ^{\frac{3}{4}} \cdot (\sqrt[3]{10})^9 = 0,0256^{\frac{3}{4}} \cdot (10^{\frac{1}{3}})^9= 0,0256^{\frac{3}{4}} \cdot 10^3 =\sqrt[4]{0,0256^3} \cdot 10^3 =\\\\ = 0,064 \cdot 1000 = 64[/tex]
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