[tex]zad. 59\\\\a)\\\\\sqrt[3]{8} =\sqrt[3]{2^{3} } =(2^{3} )^{\frac{1}{3} } =2^{3\cdot \frac{1}{3} } =2^{1} =2\\\\\sqrt{64} =\sqrt[2]{8^{2} } =(8^{2} )^{\frac{1}{2} } =8^{2\cdot \frac{1}{2} } =8^{1} =8\\\\\sqrt[3]{8}\neq \sqrt{64} ~~~~NIE\\\\b)\\\\\sqrt[3]{(-16)^{3} } =((-16)^{3} )^{\frac{1}{3} } =(-16)^{3\cdot \frac{1}{3} } =(-16)^{1} =-16\\\\-\sqrt{4^{4} } =-\sqrt[2]{4^{4} } =-(4^{4} )^{2} =-(4^{4\cdot \frac{1}{2} } )=-(4^{2} )=-16\\\\\sqrt[3]{(-16)^{3} } =-\sqrt{4^{4} } ~~~~~~TAK[/tex]
[tex]c)\\\\\sqrt{12} =\sqrt{4\cdot 3 } =\sqrt{2^{2} \cdot 3} =2\sqrt{3} \\\\\sqrt{12} =2\sqrt{3} ~~~~~~TAK\\\\d)\\\\3\cdot \sqrt{4^{2} +5^{2} } =3\cdot \sqrt{16+25} =3\cdot \sqrt{41} \\\\3\cdot \sqrt{4^{2} +5^{2} } \neq 12~~~~~~NIE[/tex]
[tex]xad.60\\\\\sqrt{5\frac{1}{3} } \cdot \sqrt{3} -2^{7} \div 2^{5} =\sqrt{(\frac{16}{3} \cdot 3)}-2^{7-5} =\sqrt{16} -2^{2} =\sqrt{4^{2} } -4=4-4=0\\\\[/tex]
korzystam ze wzorów:
[tex]\sqrt[n]{x^{n} } =(x^{n} )^{\frac{1}{n} } =x^{n\cdot \frac{1}{n} } =x^{1} =x\\\\\sqrt[n]{x} \cdot \sqrt{y} =\sqrt[n]{(x\cdot y )} \\\\x^{n} \div x^{m} =x^{n-m}[/tex]
