52

1) \begin{equation}\frac{9^{30}}{3^{59}}=\frac{\left ( 3^{2} \right )^{30}}{3^{59}}=\frac{3^{60}}{3^{59}}= \end{equation} \begin{equation}=3^{60-59}=3 \end{equation} 2) \begin{equation}\frac{8^{40}}{2^{121}}=\frac{\left ( 2^{3} \right )^{40}}{2^{121}}=\frac{2^{120}}{2^{121}}=\frac{1}{2} \end{equation} 3) \begin{equation} \frac{18^{6}}{6^{7}\cdot 3^{5}}=\frac{18^{6}}{6^{5}\cdot 6^{2}\cdot 3^{5}}= \end{equation} \begin{equation}=\frac{18^{6}}{\left ( 6^{5}\cdot 3^{5} \right )6^{2} }=\frac{\not{18^{5}}\cdot \not{18}}{\not{18^{5}}\cdot \not{36}}=\frac{1}{2} \end{equation} 4) \begin{equation}\frac{24^{12}}{6^{11}\cdot 2^{24}}=\frac{24^{12}}{6^{11}\cdot \left ( 2^{2} \right )^{12}}=\frac{24^{12}}{6^{11}\cdot 4^{12}}= \end{equation} \begin{equation}=\frac{24^{11}\cdot 24}{6^{11}\cdot 4^{11}\cdot 4}=\frac{\not{24^{11}}\cdot 24}{\not{24^{11}}\cdot 4}=6 \end{equation} 5)\begin{equation}\frac{57^{3}+57^{2}\cdot 225+171\cdot 75^{2}+75^{3}}{83^{2}+49^{2}+166\cdot 49}= \end{equation} \begin{equation}=\frac{\left ( 57+75 \right )^{3}}{\left ( 83+49 \right )^{2}}=\frac{132^{3}}{\left ( 132 \right )^{2}}=132 \end{equation} 6)\begin{equation}\frac{157^{3}-98^{3}}{157\cdot 98+98^{2}+157^{2}}= \end{equation} \begin{equation}=\frac{\left ( 157-98 \right )\left ( {157^{2}+157\cdot 98+98^{2}} \right )}{157\cdot 98+98^{2}+157^{2}}= \end{equation} \begin{equation}=157-98=59 \end{equation}

53

1)\begin{equation} \frac{b\left ( 9a-8b \right )^{2}+\left ( 3a-4b \right )^{3}}{27a^{2}}=a-b. \end{equation} \begin{equation}\frac{b\left ( 81a^{2}-144ab+64b^{2}+27a^{3}-108a^{2}b+144ab^{2}-64b^{3} \right )}{27a^{2}}= \end{equation} \begin{equation}=\frac{81a^{2}b-144ab^{2}+64b^{3}+27a^{3}-108a^{2}b+144ab^{2}-64b^{3}}{27a^{2}}= \end{equation} \begin{equation}=\frac{-27a^{2}b+27a^{3}}{27a^{2}}=\frac{27a^{2}\left ( a-b \right )}{27a^{2}}=a-b \end{equation} 2)\begin{equation}\frac{\left ( x+y \right )^{3}-y\left ( 3x+2y \right )^{2}}{3x^{2}}=\frac{x}{3}+y \end{equation} \begin{equation} \frac{x^{3}+3x^{2}y+3xy^{2}+y^{3}-y\left ( 9x^{2}+12xy+4y^{2} \right )}{3x^{2}}= \end{equation} \begin{equation}=\frac{x^{3}+3x^{2}y+3xy^{2}+y^{3}-9x^{2}y-12xy^{2}-4y^{3}}{3x^{2}}= \end{equation} \begin{equation}=\frac{x^{3}-9xy^{2}-3x^{2}y-4y^{3}}{3x^{2}}= \end{equation} \begin{equation}=\frac{-\left ( 8x^{3}+4y^{3} \right )-\left ( 9xy^{2}-3xy \right )}{3x^{2}}= \end{equation} \begin{equation}=\frac{x^{3}+3x^{2}y+3xy^{2}+y^{3}-9x^{2}y-12xy^{2}-4y^{3}}{3x^{2}}= \end{equation} \begin{equation}=\frac{x^{3}-3y^{3}-6x^{2}y-9xy^{2}}{3x^{2}} \end{equation}