Сандық өрнектерді тепе-тең түрлендіру (Рустюмова 1.1.3)

Шешуі: $${\displaystyle \frac{7^8\cdot {\left(7^2\right)}^{-2}\cdot 5^4+7^2\cdot 5^3\cdot {\left(5^{-1}\right)}^{-1}}{7^4\cdot 5^4\cdot 7^{-3}}}={\displaystyle \frac{7^8\cdot 7^{-4}\cdot 5^4+7^2\cdot 5^3\cdot 5}{7^{4-3}\cdot 5^4}}=$$ $$={\displaystyle \frac{7^4\cdot 5^4+7^2\cdot 5^4}{7\cdot 5^4}}={\displaystyle \frac{5^4\cdot 7^2\left(7^2+1\right)}{7\cdot 5^4}}={\displaystyle \frac{7\cdot 50}{1}}=350$$

Шешуі: $$={\displaystyle \frac{2^{19}(3\cdot 2+7)\cdot 52}{{13}^2\cdot {\left(2^3\right)}^8}}={\displaystyle \frac{2^{19}\cdot 13\cdot (13\cdot 4)}{{13}^2\cdot 2^{24}}}=$$ $$={\displaystyle \frac{2^{19}\cdot {13}^2\cdot 2^2}{{13}^2\cdot 2^{24}}}={\displaystyle \frac{2^{19+2-24}}{1}}=2^{-3}={\displaystyle \frac{1}{2^3}}={\displaystyle \frac{1}{8}}$$

Шешуі: $$=\frac{3^{\frac{1}{3}}\cdot {\left(3^2\right)}^{\frac{1}{4}}\cdot {\left(3^4\right)}^{\frac{1}{6}}}{3^{\frac{1}{2}}}=\frac{3^{\frac{1}{3}}\cdot 3^{\frac{1}{2}}\cdot 3^{\frac{2}{3}}}{3^{\frac{1}{2}}}=3^{\frac{1}{3}+\frac{2}{3}}=3$$

Шешуі: $$=\frac{\sqrt[8]{3^3}\cdot \sqrt[3]{8\cdot 5}\cdot \sqrt[4]{2^2}\cdot \sqrt[8]{3^5}}{\sqrt[6]{5^2}\cdot \sqrt{2}}=\frac{3^{\frac{3}{8}}\cdot 2\cdot 5^{\frac{1}{3}}\cdot 2^{\frac{1}{2}}\cdot 3^{\frac{5}{8}}}{5^{\frac{1}{3}}\cdot 2^{\frac{1}{2}}}=$$ $$=\frac{3^{\frac{3}{8}+\frac{5}{8}}\cdot 2}{1}=\frac{3\cdot 2}{1}=6$$

Шешуі: $$={\displaystyle \frac{(8+\sqrt{4\cdot 14})\cdot 1,75}{\sqrt{2}\cdot (\sqrt{7}+\sqrt{4\cdot 2})}}={\displaystyle \frac{(14+3,5\cdot \sqrt{14})(\sqrt{14}-4)}{(\sqrt{14}+4)(\sqrt{14}-4)}}=$$ $$={\displaystyle \frac{14\sqrt{14}-56+49-14\sqrt{14}}{14-16}}={\displaystyle \frac{-7}{-2}}={\displaystyle \frac{7}{2}}=3,5$$

Шешуі: $$={\displaystyle \frac{(\sqrt{9\cdot 5}-\sqrt{4\cdot 5})\cdot (\sqrt{4\cdot 3}+\sqrt{25\cdot 3})\cdot 7\sqrt{3}}{\sqrt{5}+\sqrt{36\cdot 5}}}=$$ $$={\displaystyle \frac{(3\sqrt{5}-2\sqrt{5})\cdot (2\sqrt{3}+5\sqrt{3})\cdot 7\sqrt{3}}{\sqrt{5}+6\sqrt{5}}}={\displaystyle \frac{\sqrt{5}\cdot 7\sqrt{3}\cdot 7\sqrt{3}}{7\sqrt{5}}}=$$ $$=7\cdot 3=21$$

Шешуі: $$={\displaystyle \frac{4-4\sqrt{5}+5}{3^2-{(2\sqrt[4]{5})}^2}}={\displaystyle \frac{9-4\sqrt{5}}{9-\sqrt{5}}}=1$$

Шешуі: $$={\displaystyle \frac{1-2\cdot \sqrt[4]{5}+\sqrt{5}}{3-2\cdot \sqrt{3}\cdot \sqrt[4]{3^2\cdot 5}+\sqrt{9^2\cdot 5}}}={\displaystyle \frac{1-2\cdot \sqrt[4]{5}+\sqrt{5}}{3-2\cdot 3\cdot \sqrt[4]{5}+3\sqrt{5}}}=$$ $${\displaystyle \frac{1-2\sqrt[4]{5}+\sqrt{5}}{3(1-2\sqrt[4]{5}+\sqrt{5})}}={\displaystyle \frac{1}{3}}$$

Шешуі: $$={\displaystyle \frac{(\sqrt[3]{5}-\sqrt[3]{9}){\left(\sqrt{2^5}-\sqrt{2^3}\right)}^2}{\sqrt[3]{2^3\cdot 5}-\sqrt[3]{9\cdot 2^3}}}={\displaystyle \frac{(\sqrt[3]{5}-\sqrt[3]{9})\cdot \left(32-2\sqrt{2^8}+8\right)}{2\sqrt[3]{5}-2\sqrt[3]{9}}}=$$ $$={\displaystyle \frac{(\sqrt[3]{5}-\sqrt[3]{9})\left(40-2\cdot 2^4\right)}{2(\sqrt[3]{5}-\sqrt[3]{9})}}={\displaystyle \frac{40-32}{2}}=4$$

Шешуі: $$={\displaystyle \frac{(\sqrt{3}-\sqrt{2})\cdot \sqrt{36\cdot 2}}{3\cdot (2\sqrt{6}-4)\cdot \left(\sqrt[3]{2^6}+1\right)}}={\displaystyle \frac{(\sqrt{3}-\sqrt{2})\cdot 6\sqrt{2}}{3\cdot 2(\sqrt{6}-2)\cdot (4+1)}}=$$ $$={\displaystyle \frac{6\sqrt{6}-12}{30(\sqrt{6}-2)}}={\displaystyle \frac{6(\sqrt{6}-2)}{30(\sqrt{6}-2)}}={\displaystyle \frac{1}{5}}=0,2$$

Шешуі: $$={\displaystyle \frac{\sqrt{3}(\sqrt{2}+1)-(\sqrt{2}+1)}{\sqrt{3}(\sqrt{2}+2)-(\sqrt{2}+2)}}={\displaystyle \frac{(\sqrt{2}+1)(\sqrt{3}-1)}{(\sqrt{2}+2)(\sqrt{3}-1)}}=$$ $$={\displaystyle \frac{(\sqrt{2}+1)(\sqrt{2}-2)}{(\sqrt{2}+2)(\sqrt{2}-2)}}={\displaystyle \frac{2-2\sqrt{2}+\sqrt{2}-2}{2-4}}={\displaystyle \frac{-\sqrt{2}}{-2}}={\displaystyle \frac{\sqrt{2}}{2}}$$

Шешуі: $$={\displaystyle \frac{(\sqrt{5}-\sqrt{11})\cdot (\sqrt{3\cdot 11}+\sqrt{5\cdot 3}-\sqrt{2\cdot 11}-\sqrt{5\cdot 2})}{\sqrt{25\cdot 3}-\sqrt{25\cdot 2}}}=$$ $$={\displaystyle \frac{(\sqrt{5}-\sqrt{11})\cdot \left(\sqrt{3}(\sqrt{11}+\sqrt{5})-\sqrt{2}(\sqrt{11}+\sqrt{5})\right)}{5\sqrt{3}-5\sqrt{2}}}=$$ $$={\displaystyle \frac{(\sqrt{5}-\sqrt{11})(\sqrt{3}-\sqrt{2})(\sqrt{11}+\sqrt{5})}{5(\sqrt{3}-\sqrt{2})}}={\displaystyle \frac{5-11}{5}}={\displaystyle \frac{-6}{5}}=-1,2$$

Шешуі: $$={\displaystyle \frac{\sqrt[3]{81}+2\cdot \sqrt[3]{9}\cdot \sqrt{3}+3}{\sqrt[3]{3}+2\sqrt[6]{3}+1}}={\displaystyle \frac{3\sqrt[3]{3}+2\sqrt[6]{9^2}\cdot \sqrt[6]{3^3}+3}{\sqrt[3]{3}+1+2\sqrt[6]{3}}}=$$ $$={\displaystyle \frac{3\sqrt[3]{3}+2\cdot \sqrt[6]{3^7}+3}{\sqrt[3]{3}+1+2\sqrt[6]{3}}}={\displaystyle \frac{3\sqrt[3]{3}+6\sqrt[6]{3}+3}{\sqrt[3]{3}+1+2\sqrt[6]{3}}}=$$ $$={\displaystyle \frac{(3\sqrt[3]{3}+6\sqrt[6]{3}+3)(\sqrt[3]{3}+1-2\sqrt[6]{3})}{(\sqrt[3]{3}+1+2\sqrt[6]{3})(\sqrt[3]{3}+1-2\sqrt[6]{3})}}=$$ $$={\displaystyle \frac{3\sqrt[3]{9}+3\sqrt[3]{3}-6\sqrt[3]{3}\cdot \sqrt[6]{3}+6\sqrt[6]{3}\cdot \sqrt[3]{3}+6\sqrt[6]{3}-12\sqrt[6]{9}+3\sqrt[3]{3}+3-6\sqrt[6]{3}}{{(\sqrt[3]{3}+1)}^2-4\sqrt[3]{3}}}=$$ $$={\displaystyle \frac{3\sqrt[3]{9}+3\sqrt[3]{3}-12\sqrt[6]{3^2}+3\sqrt[3]{3}+3}{\sqrt[3]{9}+2\sqrt[3]{3}+1-4\sqrt[3]{3}}}=$$ $$={\displaystyle \frac{3\sqrt[3]{9}+3\sqrt[3]{3}-12\sqrt[3]{3}+3\sqrt[3]{3}+3}{\sqrt[3]{9}-2\sqrt[3]{3}+1}}=$$ $$={\displaystyle \frac{3\sqrt[3]{9}-6\sqrt[3]{3}+3}{\sqrt[3]{9}-2\sqrt[3]{3}+1}}={\displaystyle \frac{3(\sqrt[3]{9}-2\sqrt[3]{3}+1)}{\sqrt[3]{9}-2\sqrt[3]{3}+1}}=3$$

Шешуі: $$={\displaystyle \frac{{25}^k(25-1)}{5^{2k}+2\cdot 5^k\cdot 5^{k-1}+5^{2k-2}}}={\displaystyle \frac{5^{2k}\cdot 24}{5^{2k}+2\cdot 5^{2k-1}+5^{2k-2}}}=$$ $$={\displaystyle \frac{5^{2k}\cdot 24}{5^{2k}\left(1+2\cdot 5^{-1}+5^{-2}\right)}}={\displaystyle \frac{24}{1+{\displaystyle \frac{2}{5}}+{\displaystyle \frac{1}{25}}}}={\displaystyle \frac{24}{{\displaystyle \frac{36}{25}}}}=$$ $$=24\cdot {\displaystyle \frac{25}{36}}={\displaystyle \frac{50}{3}}=16{\displaystyle \frac{2}{3}}$$

Шешуі: $$={\displaystyle \frac{3^{2k}+2\cdot 3^k\cdot 3^{k-1}+3^{2k-2}}{3^{2k-2}}}={\displaystyle \frac{3^{2k}+2\cdot 3^{2k}\cdot 3^{-1}+3^{2k}\cdot 3^{-2}}{3^{2k}\cdot 3^{-2}}}=$$ $$={\displaystyle \frac{3^{2k}\left({\displaystyle \frac{3}{1}}+{\displaystyle \frac{2}{3}}+{\displaystyle \frac{1}{9}}\right)}{3^{2k}\cdot {\displaystyle \frac{1}{9}}}}={\displaystyle \frac{{\displaystyle \frac{16}{9}}}{{\displaystyle \frac{1}{9}}}}={\displaystyle \frac{16}{9}}\cdot {\displaystyle \frac{9}{1}}=16$$