- Nếu \(a=c=0\Rightarrow\left(\frac{a-b}{c-d}\right)^{2019}=\left(\frac{b}{d}\right)^{2019}=\frac{b^{2019}}{d^{2019}}\)
\(\frac{2a^{2019}-b^{2019}}{2c^{2019}-d^{2019}}=\frac{-b^{2019}}{-d^{2019}}=\frac{b^{2019}}{d^{2019}}\Rightarrow\left(\frac{a-b}{c-d}\right)^{2019}=\frac{2a^{2019}-b^{2019}}{2c^{2019}-d^{2019}}\)
- Nếu \(a;c\ne0\)
\(\frac{a}{b}=\frac{c}{d}\Rightarrow\frac{a}{c}=\frac{b}{d}=\frac{a-b}{c-d}\)
\(\Rightarrow\frac{2a^{2019}}{2c^{2019}}=\frac{a^{2019}}{c^{2019}}=\frac{b^{2019}}{d^{2019}}=\left(\frac{a-c}{b-d}\right)^{2019}=\frac{2a^{2019}-b^{2019}}{2c^{2019}-d^{2019}}\)
Có \(\frac{a}{b}=\frac{c}{d}\Rightarrow\frac{a}{c}=\frac{b}{d}\)
Áp dụng tính chất của dãy tỉ số bằng nhau, ta có:
\(\frac{a}{c}=\frac{b}{d}=\frac{a-b}{c-d}\)
\(\Rightarrow\frac{a^{2019}}{c^{2019}}=\frac{b^{2019}}{d^{2019}}=\frac{\left(a-b\right)^{2019}}{\left(c-d\right)^{2019}}\left(1\right)\)
Có \(\frac{a}{c}=\frac{b}{d}\Rightarrow\frac{a^{2019}}{c^{2019}}=\frac{b^{2019}}{d^{2019}}\)
Áp dụng tính chất của dãy tỉ số bằng nhau, ta có:
\(\frac{a^{2019}}{c^{2019}}=\frac{b^{2019}}{d^{2019}}=\frac{2a^{2019}-b^{2019}}{2c^{2019}-d^{2019}}\left(2\right)\)
\(\left(1\right)\left(2\right)\Rightarrow\frac{\left(a-b\right)^{2019}}{\left(c-d\right)^{2019}}=\frac{2a^{2019}-b^{2019}}{2c^{2019}-d^{2019}}\)