a) \(x\left(x-y\right)+y\left(x-y\right)=\left(x+y\right)\left(x-y\right)=x^2-y^2\)

b) \(x^{n-1}\left(x+y\right)-y\left(x^{n-1}+y^{n-1}\right)=x^n+x^{n-1}y-x^{n-1}y-y^n=x^n-y^n\)

a) \(3x-7=2^3\Rightarrow3x-7=8\Rightarrow3x=15\Rightarrow x=5\)

b) \(5x+27=27\Rightarrow5x=0\Rightarrow x=0\)

c) \(40+\left(2x-3\right)=43\Rightarrow2x-3=3\Rightarrow2x=6\Rightarrow x=3\)

d) \(x:12=5\Rightarrow x=5.12=60\)

e) \(90-2\left(3x+1\right)=7.10\Rightarrow2\left(3x+1\right)=90-70\)

\(\Rightarrow2\left(3x+1\right)=20\Rightarrow3x+1=10\Rightarrow3x=9\Rightarrow x=3\)

f) \(3.2^x-15=3^2\Rightarrow3.2^x=9+15\Rightarrow3.2^x=24\Rightarrow2^x=8\)

\(\Rightarrow2^x=2^3\Rightarrow x=3\)

\(2y-x-\left\{2x-y-\left[y+3x-\left(5y-x\right)\right]\right\}\\ =2y-x-\left\{2x-y-\left(y+3x-5y+x\right)\right\}\\ =2y-x-\left(2x-y-y-3x+5y-x\right)\\ =2y-x-2x+y+y+3x-5y+x\\ =x-y\\ =\left(a+b\right)^2-\left(a-b\right)^2\\ =\left(a+b+a-b\right)\left(a+b-a+b\right)\\ =4ab\)

b) \(\sin2x+\sqrt{3}\cos2x=2\sin\left(x+\dfrac{\pi}{6}\right)\)

\(\Rightarrow\dfrac{1}{2}\sin2x+\dfrac{\sqrt{3}}{2}\cos2x=\sin\left(x+\dfrac{\pi}{6}\right)\\ \Rightarrow\sin\left(2x+\dfrac{\pi}{3}\right)=\sin\left(x+\dfrac{\pi}{6}\right)\\ \Rightarrow\left[{}\begin{matrix}2x+\dfrac{\pi}{3}=x+\dfrac{\pi}{6}+k2\pi\\2x+\dfrac{\pi}{3}=-x+\dfrac{5\pi}{6}+k2\pi\end{matrix}\right.\\ \Rightarrow\left[{}\begin{matrix}x=-\dfrac{\pi}{6}+k2\pi\\x=\dfrac{\pi}{6}+\dfrac{k2\pi}{3}\end{matrix}\right.\left(k\in Z\right)\)

a) \(\sin x+\cos x=\sqrt{2}\sin\dfrac{\pi}{12}\)

\(\Rightarrow\sqrt{2}\sin\left(x+\dfrac{\pi}{4}\right)=\sqrt{2}\sin\dfrac{\pi}{12}\\ \Rightarrow\left[{}\begin{matrix}x+\dfrac{\pi}{4}=\dfrac{\pi}{12}+k2\pi\\x+\dfrac{\pi}{4}=\dfrac{11\pi}{12}+k2\pi\end{matrix}\right.\\ \Rightarrow\left[{}\begin{matrix}x=-\dfrac{\pi}{6}+k2\pi\\x=\dfrac{2\pi}{3}+k2\pi\end{matrix}\right.\left(k\in Z\right)\)