Сандық өрнектерді тепе-тең түрлендіру (Рустюмова 1.1.9)

Шешуі: $$=\left(\left(4\sqrt{6}+\sqrt{39}\right)+\left(2\sqrt{26}+6\right)\right)\cdot \left(\left(4\sqrt{6}+\sqrt{39}\right)-\left(2\sqrt{26}+6\right)\right)=$$ $$={\left(4\sqrt{6}+\sqrt{39}\right)}^2-{\left(2\sqrt{26}+6\right)}^2=$$ $$=96+39+8\sqrt{6\cdot 39}-\left(4\cdot 26+36+24\sqrt{26}\right)=$$ $$=135+8\sqrt{26\cdot 9}-140-24\sqrt{26}=-5+24\sqrt{26}-24\sqrt{26}=-5.$$ Жауабы: $-5$

Шешуі: $$=\left(2\sqrt{7}-2\sqrt{3}\right)\cdot \sqrt{{\left(\sqrt{3}+\sqrt{7}\right)}^2}=\left(2\sqrt{7}-2\sqrt{3}\right)\cdot \left(\sqrt{3}+\sqrt{7}\right)=$$ $$=2\cdot \left(\sqrt{7}-\sqrt{3}\right)\cdot \left(\sqrt{7}+\sqrt{3}\right)=2\cdot \left(7-3\right)=8$$ Жауабы: $8$

Шешуі: $$={\displaystyle \frac{\sqrt{{\left(7-\sqrt{48}\right)}^2}+4\sqrt{3}}{\sqrt{{\left(5-\sqrt{27}\right)}^2-3\sqrt{3}}}}\cdot \left(\sqrt{{\left(1+\sqrt{2}\right)}^2}+\sqrt{{\left(1-\sqrt{2}\right)}^2}\right)=$$ $$={\displaystyle \frac{7-4\sqrt{3}+4\sqrt{3}}{3\sqrt{3}-5-3\sqrt{3}}}\cdot (1+\sqrt{2}+\sqrt{2}-1)={\displaystyle \frac{7}{-5}}\cdot 2\sqrt{2}=-{\displaystyle \frac{14\sqrt{2}}{5}}$$ Жауабы: $-{\displaystyle \frac{14\sqrt{2}}{5}}$

Шешуі: $$=\left(15\cdot {\displaystyle \frac{1}{16}}+{\displaystyle \frac{{\displaystyle \frac{1}{16}}\cdot \sqrt{11}}{\sqrt[4]{121}}}\right)\cdot {\left({\displaystyle \frac{(1+\sqrt[4]{9})(\sqrt{3}-1)}{4}}\right)}^{-1}=$$ $$=\left({\displaystyle \frac{15}{16}}+{\displaystyle \frac{\sqrt{11}}{16\sqrt{11}}}\right)\cdot {\left({\displaystyle \frac{(1+\sqrt{3})(\sqrt{3}-1)}{4}}\right)}^{-1}=\left({\displaystyle \frac{15}{16}}+{\displaystyle \frac{1}{16}}\right)\cdot 2=2$$ Жауабы: $2$

Шешуі: $$=(2\sqrt{6}-5{)}^2-10\sqrt{{(2\sqrt{6}-5)}^2}+1=49-20\sqrt{6}-10\cdot \left|2\sqrt{6}-5\right|+1=$$ $$=49-20\sqrt{6}-10\cdot (5-2\sqrt{6})+1=50-20\sqrt{6}-50+20\sqrt{6}=0$$ Жауабы: $0$

Шешуі: $$=\sqrt{6}+\sqrt{5}-{\displaystyle \frac{\sqrt{11+2\sqrt{30}}}{\sqrt{121-120}}}=$$ $$=\sqrt{6}+\sqrt{5}-{\displaystyle \frac{\sqrt{{(\sqrt{6}+\sqrt{5})}^2}}{1}}=\sqrt{6}+\sqrt{5}-\left|\sqrt{6}+\sqrt{5}\right|=$$ $$=\sqrt{6}+\sqrt{5}-(\sqrt{6}+\sqrt{5})=\sqrt{6}+\sqrt{5}-\sqrt{6}-\sqrt{5}=0$$ Жауабы: $0$

Шешуі: $$=\sqrt[6]{{(5\sqrt{2}-7)}^2}\cdot \sqrt[6]{{(3+2\sqrt{2})}^3}=$$ $$=\sqrt[6]{50-14\cdot 5\sqrt{2}+49}\cdot \sqrt[6]{27+3\cdot 18\sqrt{2}+72+16\sqrt{2}}=$$ $$=\sqrt[6]{9801-9800}=1.$$ Жауабы: $1$

Шешуі: $$={\displaystyle \frac{\sqrt{16+2\cdot 4\cdot \sqrt{5}+5}}{\sqrt{5}+4}}\cdot \sqrt{{(2-\sqrt{5})}^2}=$$ $$={\displaystyle \frac{\sqrt{{(4+\sqrt{5})}^2}}{4+\sqrt{5}}}\cdot \left|2-\sqrt{5}\right|={\displaystyle \frac{\left|4+\sqrt{5}\right|}{4+\sqrt{5}}}\cdot (\sqrt{5}-2)=$$ $$={\displaystyle \frac{4+\sqrt{5}}{4+\sqrt{5}}}(\sqrt{5}-2)=\sqrt{5}-2.$$ Жауабы: $\sqrt{5}-2$

Шешуі: $$={\displaystyle \frac{\sqrt{6,3\cdot 1,7}\cdot {\displaystyle \frac{6,3-1,7}{\sqrt{6,3\cdot 1,7}}}}{\sqrt{{6,3}^2+2\cdot 6,3\cdot 1,7+{1,7}^2-4\cdot 6,3\cdot 1,7}}}=$$ $$={\displaystyle \frac{6,3-1,7}{\sqrt{{\left(6,3-1,7\right)}^2}}}={\displaystyle \frac{4,6}{4,6}}=1$$ Жауабы: $1$

Шешуі: $$=\left({\displaystyle \frac{\sqrt{102\cdot 1020}}{{\displaystyle \frac{30}{7}}\left(0,15+{\displaystyle \frac{3}{20}}\right)}}+4\sqrt{10}\right)\cdot {\displaystyle \frac{3}{\sqrt{40}}}=$$ $$=\left({\displaystyle \frac{2\cdot 7\cdot 17\sqrt{10}+12\sqrt{10}}{3}}\right)\cdot {\displaystyle \frac{3}{\sqrt{40}}}={\displaystyle \frac{250\sqrt{10}}{\sqrt{40}}}=125$$ Жауабы: $125$

Шешуі: $$={\left({\displaystyle \frac{2(\sqrt{7}-\sqrt{5})}{2}}+{\displaystyle \frac{3(\sqrt{5}+\sqrt{2})}{3}}\right)}^{-2}:{\left(1+\sqrt{{\displaystyle \frac{7}{2}}}\right)}^{-2}=$$ $$={\left(\sqrt{7}-\sqrt{5}+\sqrt{5}+\sqrt{2}\right)}^{-2}:{\left({\displaystyle \frac{\sqrt{2}+\sqrt{7}}{\sqrt{2}}}\right)}^{-2}=$$ $$={\left(\sqrt{7}+\sqrt{2}\right)}^{-2}:{\left({\displaystyle \frac{\sqrt{7}+\sqrt{2}}{\sqrt{2}}}\right)}^{-2}={\left({\displaystyle \frac{\left(\sqrt{7}+\sqrt{2}\right)\cdot \sqrt{2}}{\left(\sqrt{7}+\sqrt{2}\right)}}\right)}^{-2}=$$ $$={\left(\sqrt{2}\right)}^{-2}={\left({\displaystyle \frac{1}{\sqrt{2}}}\right)}^2={\displaystyle \frac{1}{2}}$$ Жауабы: ${\displaystyle \frac{1}{2}}$

Шешуі: $$=\left(\sqrt[6]{{\left(2+\sqrt{5}\right)}^2}+\sqrt[3]{2+\sqrt{5}}\right)\cdot \sqrt[3]{\sqrt{5}-2}=$$ $$=\left(\sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2+\sqrt{5}}\right)\cdot \sqrt[3]{\sqrt{5}-2}=$$ $$=2\sqrt[3]{2+\sqrt{5}}\cdot \sqrt[3]{\sqrt{5}-2}=2\sqrt{5-4}=2\cdot 1=2.$$ Жауабы: $2$

Шешуі: $$={\left(1+\sqrt{2}\right)}^2+{\left(3+2\sqrt{2}\right)}^{-1}=$$ $$1+2\sqrt{2}+2+{\displaystyle \frac{1}{3+2\sqrt{2}}}=3+2\sqrt{2}+{\displaystyle \frac{3-2\sqrt{2}}{9-8}}=$$ $$=3+2\sqrt{2}+3-2\sqrt{2}=6$$ Жауабы: $6$

Шешуі: $$=\left(\sqrt{3}-2\sqrt[4]{81}+\sqrt{27}+7\right)\cdot \left(\sqrt{3}+2\sqrt[4]{81}+\sqrt{27}-7\right)=$$ $$=(4\sqrt{3}+1)\cdot (4\sqrt{3}-1)=48-1=47$$ Жауабы: $47$

Шешуі: $$=\sqrt{13+30\sqrt{2+\sqrt{{\left(1+\sqrt{8}\right)}^2}}}=\sqrt{13+30\sqrt{2+1+\sqrt{8}}}=$$ $$=\sqrt{13+30\sqrt{2+1+2\sqrt{2}}}=\sqrt{13+30\sqrt{{\left(\sqrt{2}\right)}^2+2\cdot \sqrt{2}\cdot 1+1^2}}=$$ $$=\sqrt{13+30\sqrt{{\left(1+\sqrt{2}\right)}^2}}=\sqrt{13+30\cdot \left(1+\sqrt{2}\right)}=$$ $$=\sqrt{13+30+30\sqrt{2}}=\sqrt{43+30\sqrt{2}}=$$ $$=\sqrt{{(5+3\sqrt{2})}^2}=5+3\sqrt{2}$$ Жауабы: $5+3\sqrt{2}$

Шешуі: $$=\sqrt[3]{(\sqrt[4]{52}+\sqrt{5})(\sqrt[4]{52}-\sqrt{5})(2\sqrt{13}+5)}=$$ $$=\sqrt[3]{(\sqrt{52}-5)(\sqrt{52}+5)}=\sqrt[3]{52-25}=\sqrt[3]{27}=3$$ Жауабы: $3$

Шешуі: $$={\displaystyle \frac{\sqrt{{(1-\sqrt{11})}^2}}{\sqrt{{(3-2\sqrt{2})}^2}}}-{\displaystyle \frac{\sqrt{{(1+\sqrt{11})}^2}}{\sqrt{{(3+2\sqrt{2})}^2}}}={\displaystyle \frac{\sqrt{11}-1}{3-2\sqrt{2}}}-{\displaystyle \frac{1+\sqrt{11}}{3+2\sqrt{2}}}=$$ $$=(\sqrt{11}-1)(3+2\sqrt{2})-(1+\sqrt{11})(3-2\sqrt{2})=$$ $$=3\sqrt{11}-3+2\sqrt{22}-2\sqrt{2}-3-3\sqrt{11}+2\sqrt{2}+2\sqrt{22}=4\sqrt{22}-6$$ Жауабы: $4\sqrt{22}-6$

Шешуі: $$={7521}^2-\left(7521+1\right)\left(7521-1\right)=$$ $$={7521}^2-\left({7521}^2-1^2\right)=1$$ Жауабы: $1$

Шешуі: $$=\left|\sqrt{5}-3\right|-3\sqrt{5}+7=3-\sqrt{5}-3\sqrt{5}+7=10-4\sqrt{5}$$ Жауабы: $10-4\sqrt{5}$

Шешуі: $$=\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{2}{5}}\right)+\left({\displaystyle \frac{2}{5}}-{\displaystyle \frac{3}{8}}\right)+\left({\displaystyle \frac{3}{8}}-{\displaystyle \frac{4}{11}}\right)+\left({\displaystyle \frac{4}{11}}-{\displaystyle \frac{5}{14}}\right)+\left({\displaystyle \frac{5}{14}}-{\displaystyle \frac{6}{17}}\right)=$$ $$={\displaystyle \frac{1}{2}}-{\displaystyle \frac{2}{5}}+{\displaystyle \frac{2}{5}}-{\displaystyle \frac{3}{8}}+{\displaystyle \frac{3}{8}}-{\displaystyle \frac{4}{11}}+{\displaystyle \frac{4}{11}}-{\displaystyle \frac{5}{14}}+{\displaystyle \frac{5}{14}}-{\displaystyle \frac{6}{17}}=$$ $$={\displaystyle \frac{1}{2}}-{\displaystyle \frac{6}{17}}={\displaystyle \frac{5}{34}}$$ Жауабы: ${\displaystyle \frac{5}{34}}$