Алгебра
Свойства корня n-ой степени
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1)
![displaystyle sqrt[3]{ frac{27a^5}{b^2}}* sqrt[3]{ frac{ab^{11}}{8}}= sqrt[3]{ frac{3^3*a^5*a*b^{11}}{b^2*2^3}}= sqrt[3]{ frac{3^3*(a^2)^3*(b^3)^3}{2^3}}=\\= frac{3a^2b^3}{2}](https://tex.z-dn.net/?f=displaystyle++sqrt%5B3%5D%7B+frac%7B27a%5E5%7D%7Bb%5E2%7D%7D%2A+sqrt%5B3%5D%7B+frac%7Bab%5E%7B11%7D%7D%7B8%7D%7D%3D+sqrt%5B3%5D%7B+frac%7B3%5E3%2Aa%5E5%2Aa%2Ab%5E%7B11%7D%7D%7Bb%5E2%2A2%5E3%7D%7D%3D+sqrt%5B3%5D%7B+frac%7B3%5E3%2A%28a%5E2%29%5E3%2A%28b%5E3%29%5E3%7D%7B2%5E3%7D%7D%3D%5C%5C%3D+frac%7B3a%5E2b%5E3%7D%7B2%7D+++++ "displaystyle sqrt[3]{ frac{27a^5}{b^2}}* sqrt[3]{ frac{ab^{11}}{8}}= sqrt[3]{ frac{3^3*a^5*a*b^{11}}{b^2*2^3}}= sqrt[3]{ frac{3^3*(a^2)^3*(b^3)^3}{2^3}}=\\= frac{3a^2b^3}{2}")
2)
![displaystyle sqrt[4]{b}: sqrt{b^3}* sqrt{ sqrt{b^{13}}}=b^{1/4}:b^{3/2}* sqrt{b^{13/2}}=\\=b^{1/4}:b^{3/2}*b^{13/4}=b^{1/4-3/2+13/4}=b^{2}](https://tex.z-dn.net/?f=displaystyle++sqrt%5B4%5D%7Bb%7D%3A+sqrt%7Bb%5E3%7D%2A+sqrt%7B+sqrt%7Bb%5E%7B13%7D%7D%7D%3Db%5E%7B1%2F4%7D%3Ab%5E%7B3%2F2%7D%2A+sqrt%7Bb%5E%7B13%2F2%7D%7D%3D%5C%5C%3Db%5E%7B1%2F4%7D%3Ab%5E%7B3%2F2%7D%2Ab%5E%7B13%2F4%7D%3Db%5E%7B1%2F4-3%2F2%2B13%2F4%7D%3Db%5E%7B2%7D++++ "displaystyle sqrt[4]{b}: sqrt{b^3}* sqrt{ sqrt{b^{13}}}=b^{1/4}:b^{3/2}* sqrt{b^{13/2}}=\\=b^{1/4}:b^{3/2}*b^{13/4}=b^{1/4-3/2+13/4}=b^{2}")
3)
![displaystyle sqrt[3]{x}+ sqrt[6]{x}-12=0\\ sqrt[6]{x}=t\\t^2+t-12=0\D=1+48=49=7^2\\t_{1.2}= frac{-1pm 7}{2}; t_1=-4; t_2=3\\ sqrt[6]{x}=3; x=3^6\\ sqrt[6]{x} neq -4\\x= 729](https://tex.z-dn.net/?f=displaystyle++sqrt%5B3%5D%7Bx%7D%2B+sqrt%5B6%5D%7Bx%7D-12%3D0%5C%5C+sqrt%5B6%5D%7Bx%7D%3Dt%5C%5Ct%5E2%2Bt-12%3D0%5CD%3D1%2B48%3D49%3D7%5E2%5C%5Ct_%7B1.2%7D%3D+frac%7B-1pm+7%7D%7B2%7D%3B+t_1%3D-4%3B+t_2%3D3%5C%5C+sqrt%5B6%5D%7Bx%7D%3D3%3B+x%3D3%5E6%5C%5C+sqrt%5B6%5D%7Bx%7D+neq+-4%5C%5Cx%3D+729+++++ "displaystyle sqrt[3]{x}+ sqrt[6]{x}-12=0\\ sqrt[6]{x}=t\\t^2+t-12=0\D=1+48=49=7^2\\t_{1.2}= frac{-1pm 7}{2}; t_1=-4; t_2=3\\ sqrt[6]{x}=3; x=3^6\\ sqrt[6]{x} neq -4\\x= 729")
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