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4 6 . 9 5 + 6 9 . 120 8 4 . 3 12 − 6 11 = 2 2 6 . 3 2 5 + 2 9 .3 9 .2 3 .3.5 2 3 4 .3 12 − 2 11 .3 11 = 2 12 .3 10 + 2 12 .3 10 .5 2 12 .3 12 − 2 11 .3 11 = 2 12 .3 10 1 + 5 2 11 .3 11 2.3 − 1 = 2.6 3.5 = 4 5
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\(A=\frac{4^6.9^5+69.120}{8^4.3^{12}+6^{11}}=\frac{2^{12}.3^{10}+2^3.3^2.115}{2^{12}.3^{12}+\left(2.3\right)^{11}}=\frac{3^2.2^3\left(115+2^9.3^8\right)}{6^{11}\left(6+1\right)}=\frac{115+2^9.3^8}{6^8.3.7}\)
\(B=\frac{10^4+5.10^3+5^4}{25}=\frac{\left(10^2\right)^2+2.5^2.10^2+\left(5^2\right)^2}{25}=\frac{\left(10^2+5^2\right)^2}{25}=\frac{125^2}{25}=\frac{25.625}{25}=625\)
\(C=\frac{8^{10}+4^{10}}{8^4+4^{11}}=\frac{4^{10}.2^{10}+4^{10}}{4^4.4^7+4^4.2^4}=\frac{4^{10}\left(2^{10}+1\right)}{4^4.2^4\left(2^{10}+1\right)}=\frac{4^6}{2^4}=256\)
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Gọi 3 phần đó là: x, y, z
Ta có:
\(\frac{x}{\frac{1}{3}}=\frac{y}{\frac{1}{4}}=\frac{z}{\frac{1}{5}}\) và x + y + z = 611
Áp dụng tính chất của dãy tỉ số bằng nhau, có:
\(\frac{x}{\frac{1}{3}}=\frac{y}{\frac{1}{4}}=\frac{z}{\frac{1}{5}}=\frac{x+y+z}{\frac{1}{3}+\frac{1}{4}+\frac{1}{5}}=\frac{611}{\frac{47}{60}}=780\)
\(\frac{x}{\frac{1}{3}}=780\Rightarrow x=780\cdot\frac{1}{3}=260\)
\(\frac{y}{\frac{1}{4}}=780\Rightarrow y=780\cdot\frac{1}{4}=195\)
\(\frac{z}{\frac{1}{5}}=780\Rightarrow z=780\cdot\frac{1}{5}=156\)
Vậy 3 phần đó lần lượt là: 260, 195, 156
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Gọi 3 phần là a; b; c với a+b+c=611
\(3a=4b=5c\Rightarrow\dfrac{a}{20}=\dfrac{b}{15}=\dfrac{c}{12}=\dfrac{a+b+c}{20+15+12}=\dfrac{611}{47}=13\)
\(\Rightarrow a=13.20=260\)
\(b=13.15=195\)
\(c=13.12=156\)
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\(\frac{4^6.9^5+6^9.120}{8^4.3^{12}-6^{11}}\)
\(=\frac{2^{12}.3^{10}+2^9.3^9.2^3.3.5}{2^{12}.3^{12}-2^{11}.3^{11}}\)
\(=\frac{2^{12}.3^{10}+2^{12}.3^{10}.5}{2^{11}.3^{11}.\left(2.3-1\right)}\)
\(=\frac{2^{12}.3^{10}.\left(1+5\right)}{2^{11}.3^{11}.5}\)
\(=\frac{2.6}{3.5}=\frac{4}{5}\)
\(\frac{4^6.9^5+6^9.120}{8^4.3^{12}-6^{11}}\)
\(=\frac{2^{12}.3^{10}+2^9.3^9.2^3.3.5}{2^{12}.3^{12}-6^{11}}\)
\(=\frac{2^{12}.3^{10}+2^{12}.3^{10}.5}{6^{12}-6^{11}}\)
\(=\frac{2^{12}.3^{10}\left(1+5\right)}{6^{11}.6-6^{11}}\)
\(=\frac{\left(2.3\right)^{10}.2^2.6}{6^{11}\left(6-1\right)}\)
\(=\frac{6^{10}.6.2^2}{6^{11}.5}=\frac{6^{11}.2^2}{6^{11}.5}=\frac{2^2}{5}=\frac{4}{5}\)