1) a) \(\left(x-3\right)\left(x^2+3x+9\right)-\left(54+x^3\right)\)

\(=\left(x^3-3^3\right)-\left(54+x^3\right)\\ =\left(x^3-27\right)-54-x^3\\ =-27-54\\ =-81\)

b) \(\left(3x+y\right)\left(9x^2-3xy+y^2\right)-\left(3x-y\right)\left(9x^2+3xy+y^2\right)\)

\(=\left[\left(3x\right)^3+y^3\right]-\left[\left(3x\right)^3-y^3\right]\\ =2y^3\)

2) a) \(\left(x+3y\right)\left(x^2-3xy+9y^2\right)=x^3+27y^3\)

b) \(\left(2x-3y\right)\left(4x^2+6xy+9y^2\right)=8x^3-27y^3\)

biến đổi vế phải ta có:

\(VP=\left(a+b\right)^3-3ab\left(a+b\right)\\ =\left(a^3+3a^2b+3ab^2+b^3\right)-\left(3a^2b+3ab^2\right)\\ =a^3+3a^2b+3ab^2+b^3-3a^2b-3ab^2\\ =a^3+b^3\\ =VT\)

VP = VT (đpcm)

ta có:

\(6^2.6^4-4^3\left(3^6-1\right)\\ =6^6-\left(2^2\right)^3\left(3^6-1\right)\\ =6^6-2^6\left(3^6-1\right)\\ =6^6-2^6.3^6+2^6\\ =6^6-6^6+2^6\\ =0+2^6=2^6\)