1.
[tex](11\frac19)^{-\frac12}-(-0.2)^{-3}*81^{-0.75}+(0.512)^{\frac13}=\\(\frac{100}9)^{-\frac12}-(-\frac1{5})^{-3}*(9^2)^{-\frac34}+(\frac{64}{125})^{\frac13}=\\\sqrt{\frac{9}{100}}-(-5)^3*9^{-\frac32}+\sqrt[3]{\frac{64}{125}}=\\\frac{3}{10}-(-125)*\sqrt{\frac{1}{9^3}}+\frac45=\\\frac3{10}-(-125)*\frac1{27}+\frac45=\\\frac3{10}-(-\frac{125}{27})+\frac45=\\\frac3{10}+\frac{125}{27}+\frac45=\\[/tex]
[tex]\frac3{10}+\frac{125}{27}+\frac8{10}=\\\frac{11}{10}+\frac{125}{27}=\\\frac{297}{270}+\frac{1250}{270}=\frac{1547}{270}=5\frac{197}{270}[/tex]
2.
[tex]\frac{b^{10}:b^0}{(b^4)^4*b^5}=\frac{b^{10}}{b^{16}*b^5}=b^{10-(16+5)}=b^{10-21}=b^{-11}=\frac1{b^{11}}[/tex]
3.
[tex]a)\\log_4\frac16-log_4\frac13=log_4(\frac16:\frac13)=log_4(\frac16*3)=log_42=\frac12\\b)\\8^{2+log_23}=8^2*8^{log_23}=64*(2^3)^{log_23}=64*2^{3log_23}=64*2^{log_23^3}=64*2^{log_227}=64*27=1728[/tex]
4.
[tex]3(2x-y)^2-(2y+x)^2=\\3((2x)^2-2*2x*y+y^2)-((2y)^2+2*2y*x+x^2)=\\3(4x^2-4xy+y^2)-(4y^2+4xy+x^2)=\\12x^2-12xy+3y^2-4y^2-4xy-x^2=\\11x^2-16xy-y^2=\\11*(-2)^2-16*(-2)*(-1)-(-1)^2=\\11*4-32-1=\\44-33=11[/tex]
[tex]6.\\\\a)\frac{20}{3\sqrt5}*\frac{\sqrt5}{\sqrt5}=\frac{20\sqrt5}{15}=\frac{4\sqrt5}3\\\\b) \frac{2}{2-3\sqrt3}*\frac{2+3\sqrt3}{2+3\sqrt3}=\frac{2(2+3\sqrt3)}{2^2-(3\sqrt3)^2}=\frac{4+6\sqrt3}{4-27}=\frac{4+6\sqrt3}{-23}=-\frac{4+6\sqrt3}{23}[/tex]
7.
[tex](2-3\sqrt3)^2-(1+5\sqrt2)^2=\\4-12\sqrt3+27-(1+10\sqrt2+50)=\\31-12\sqrt3-(51+10\sqrt2)=\\31-12\sqrt3-51-10\sqrt2=\\-20-12\sqrt3-10\sqrt2[/tex]
[tex]5^*. \\\\a=log_581=log_5(3^4)=4log_53 /:2\\\frac{a}2=2log_53\\\\b=log_58=log_5(2^3)=3log_52 /:3\\\frac{b}3=log_52\\\\\\log_29=\frac{log_59}{log_52}=\frac{log_5(3^2)}{log_52}=\frac{2log_53}{log_52}=\frac{\frac{a}2}{\frac{b}3}=\frac{a}2:\frac{b}3=\frac{a}2*\frac3{b}=\frac{3a}{2b}[/tex]
