Bài 8:
a: Ta có: \(A=\left(x-y\right)^2+3xy\left(x-y\right)\)
\(=x^2-2xy+y^2+3x^2y-3xy^2\)
b: Ta có: \(B=\left(x+y\right)^3+3\left(x-y\right)\left(x+y\right)^2+3\left(x-y\right)^2\left(x+y\right)+\left(x-y\right)^3\)
\(=\left(x+y+x-y\right)^3\)
\(=\left(2x\right)^3=8x^3\)
Bài 7:
a: \(A=8x^6-12x^4+6x^2-1=\left(2x^2-1\right)^3\)
b: \(B=8\left(\dfrac{1}{2}x+y\right)^3-6\left(x+2y\right)^2x+12\left(x+2y\right)\cdot x^2-8x^3\)
\(=\left(x+2y\right)^3-3\cdot\left(x+2y\right)^2\cdot2x+3\cdot\left(x+2y\right)\cdot\left(2x\right)^2-\left(2x\right)^3\)
\(=\left(x+2y-2x\right)^3\)
\(=\left(-x+2y\right)^3\)
c: Ta có: \(C=\left(x-y\right)^3-\dfrac{3}{2}\left(x-y\right)^2\cdot y+\dfrac{3}{4}\left(x-y\right)\cdot y^2-\dfrac{y^3}{8}\)
\(=\left(x-y\right)^3-3\cdot\left(x-y\right)^2\cdot\dfrac{1}{2}y+3\cdot\left(x-y\right)\cdot\left(\dfrac{1}{2}y\right)^2-\left(\dfrac{1}{2}y\right)^3\)
\(=\left(x-\dfrac{3}{2}y\right)^3\)
Bài 6:
a: \(\left(3x+1\right)^3=27x^2+27x+9x+1\)
b: \(\left(2x-\dfrac{1}{x}\right)^3=8x^3-3\cdot4x^2\cdot\dfrac{1}{x}+3\cdot2x\cdot\dfrac{1}{x^2}-\dfrac{1}{x^3}\)
\(=8x^3-12x+\dfrac{6}{x}-\dfrac{1}{x^3}\)
c: \(\left(y-\dfrac{xy}{3}\right)^3\)
\(=y^3-3\cdot y^2\cdot\dfrac{xy}{3}+3\cdot y\cdot\dfrac{x^2y^2}{9}-\dfrac{x^3y^3}{27}\)
\(=y^3-xy^3+\dfrac{1}{3}x^2y^3-\dfrac{x^3y^3}{27}\)
d: \(\left(\dfrac{1}{y^2}+\dfrac{y}{x}\right)^3\)
\(=\dfrac{1}{y^6}+3\cdot\dfrac{1}{y^4}\cdot\dfrac{y}{x}+3\cdot\dfrac{1}{y^2}\cdot\dfrac{y^2}{x^2}+\dfrac{y^3}{x^3}\)
\(=\dfrac{1}{y^6}+\dfrac{3}{y^3x}+\dfrac{3}{x^2}+\dfrac{y^3}{x^3}\)
