Rút gọn phân số sau: \(\frac{\left(2^{17}+5^{17}\right)\cdot\left(3^{14}-5^{12}\right)\cdot\left(2^4-4^2\right)}{15^2+5^3+67^7}\)
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12 tháng 8 2015
\(=\frac{\left(2^{17}+5^{17}\right)\left(3^{14}-5^{12}\right)\left(16-16\right)}{15^2+5^3+67^7}=\frac{\left(2^{17}+5^{17}\right)\left(3^{14}-5^{12}\right).0}{15^2+5^3+67^7}=0\)
21 tháng 9 2017
Theo đề ta có :
\(\frac{3}{\left(x+2\right)\left(x+5\right)}+\frac{5}{\left(x+5\right)\left(x+10\right)}+\frac{7}{\left(x+10\right)\left(x+17\right)}=\frac{x}{\left(x+2\right)\left(x+17\right)}\)
\(\Leftrightarrow\frac{\left(x+5\right)-\left(x+2\right)}{\left(x+2\right)\left(x+5\right)}+\frac{\left(x+10\right)-\left(x+5\right)}{\left(x+5\right)\left(x+10\right)}+\frac{\left(x+17\right)-\left(x+10\right)}{\left(x+10\right)\left(x+17\right)}=\frac{x}{\left(x+2\right)\left(x+17\right)}\)
\(\Rightarrow\frac{1}{x+2}-\frac{1}{x+5}+\frac{1}{x+5}-\frac{1}{x+10}+\frac{1}{x+10}-\frac{1}{x+17}=\frac{x}{\left(x+2\right)\left(x+17\right)}\)
\(\Rightarrow\frac{1}{x+2}-\frac{1}{x+17}=\frac{x}{\left(x+2\right)\left(x+17\right)}\)
\(\Rightarrow\frac{\left(x+17\right)-\left(x+2\right)}{\left(x+2\right)\left(x+17\right)}=\frac{x}{\left(x+2\right)\left(x+17\right)}\)
\(\Rightarrow\left(x+17\right)-\left(x+2\right)=x\)
\(\Rightarrow x=15\)
\(\frac{\left(2^{17}+5^{17}\right)\left(3^{14}-5^{12}\right)\left(2^4-4^2\right)}{15^2+5^3+67^7}=\frac{\left(2^{17}+5^{17}\right)\left(3^{14}-5^{12}\right).0}{15^2+5^3+67^7}=0\)