Odpowiedź:
[tex]a)~~5\cdot \sqrt{16\dfrac{1}{3} } + 3^{6} \div 3^{3} =\dfrac{35\sqrt{3} }{3} +27\\\\\\b)~~\sqrt{15} \cdot \sqrt{1\dfrac{2}{3} } =5\\\\[/tex]
[tex]c)~~10\cdot (\sqrt{45} -\sqrt{27} )=30\sqrt{5} -30\sqrt{3} \\\\d)~~(\sqrt{75} -\sqrt{50} )\div 10 = \dfrac{\sqrt{3} }{2} -\dfrac{\sqrt{2} }{2}[/tex]
[tex]e)~~4\cdot \left(\sqrt{\dfrac{1}{2} } +\sqrt{2} \right)=6\sqrt{2}[/tex]
[tex]f)~~\dfrac{3,4\cdot 10^{3} }{2\cdot 10^{5} } =0,017[/tex]
Szczegółowe wyjaśnienie:
Korzystamy ze wzorów:
[tex]\sqrt[n]{x^{n} } =x^{n\cdot \frac{1}{n} } =x^1=x\\\\\dfrac{x^{n} }{x^{m} } =x^{n} \div x^{m} =x^{n-m} \\\\\sqrt{x} \cdot \sqrt{y} =\sqrt{x\cdot y}[/tex]
Pamiętamy o kolejności wykonywanych działań:
Obliczamy:
[tex]a)\\~\\~5\cdot \sqrt{16\dfrac{1}{3} } + 3^{6} \div 3^{3} =5\cdot \sqrt{\dfrac{49}{3} } + 3^{6-3}=5\cdot \sqrt{\dfrac{7^{2} }{3} } + 3^{3}=5\cdot \dfrac{7}{\sqrt{3} } \cdot \dfrac{\sqrt{3} }{\sqrt{3} } +27=\dfrac{35\sqrt{3} }{3} +27\\\\\\b)\\~\\~\sqrt{15} \cdot \sqrt{1\dfrac{2}{3} } =\sqrt{15} \cdot \sqrt{\dfrac{5}{3} } = \sqrt{15\cdot \dfrac{5}{3} } =\sqrt{25} =\sqrt{5^{2} } =5\\\\[/tex]
[tex]c)\\~\\~10\cdot (\sqrt{45} -\sqrt{27} )=10\cdot (\sqrt{9\cdot 5} -\sqrt{9\cdot 3} )=10\cdot (\sqrt{3^2\cdot 5} -\sqrt{3^2\cdot 3} )=10\cdot (3\sqrt{ 5} -3\sqrt{3} )=30\sqrt{5} -30\sqrt{3} \\\\d)\\~\\~(\sqrt{75} -\sqrt{50} )\div 10 =(\sqrt{25\cdot 3} -\sqrt{25\cdot 2} )\div 10=(\sqrt{5^2\cdot 3} -\sqrt{5^2\cdot 2} )\div 10=(5\sqrt{3} -5\sqrt{2} )\div 10=5(\sqrt{3} -\sqrt{2} )\div 10=(\sqrt{3} -\sqrt{2} )\div 2==\dfrac{\sqrt{3}-\sqrt{2} }{2} = \dfrac{\sqrt{3} }{2} -\dfrac{\sqrt{2} }{2}[/tex]
[tex]e)\\~\\~4\cdot \left(\sqrt{\dfrac{1}{2} } +\sqrt{2} \right)=4\cdot \left(\sqrt{\dfrac{1^2}{2} } +\sqrt{2} \right)=4\cdot \left(\dfrac{1}{\sqrt{2} } +\sqrt{2} \right)=4\cdot \left(\dfrac{1+\sqrt{2}\cdot \sqrt{2} }{\sqrt{2} } \right)=4\cdot \left(\dfrac{1+2}{\sqrt{2} } \right)=4\cdot \dfrac{3}{\sqrt{2} } =\dfrac{12}{\sqrt{2} }\cdot \dfrac{\sqrt{2} }{\sqrt{2} } =\dfrac{12\sqrt{2} }{2 }=6\sqrt{2}[/tex]
[tex]f)\\~\\~\dfrac{3,4\cdot 10^{3} }{2\cdot 10^{5} } =(3,4\div 2 )\cdot 10^{3-5} =1,7\cdot 10^{-2} =1\dfrac{7}{10} \cdot \left(\dfrac{1}{10} \right)^{2} =\dfrac{17}{10}\cdot \dfrac{1}{100}=\dfrac{17}{1000}=0,017[/tex]
Odpowiedź:
Szczegółowe wyjaśnienie:
5√16 1/3+3^6:3^3=5√49/3+3^3=5*7/√3+27=35/√3+27=35√3/3+27=11 2/3√3+27
√15*√1 2/3=√15*√5/3=√75/3=√25=5
10(√45-√27)=10(3√5-3√3)=10*3√5-10*3√3=300√5-30√3=30(√5-√3)
(√75-√50)/10=(5√5- 5√2):10=5√5/10- 5√2/10=√5/2- √2/2
4(√1/2+√2)=4*√1/2+4√2=√16*√1/2+4√2=√16/2+4√2=√8+4√2=2√2+4√2=6√2
