a: \(\left(2x^2+\frac13\right)^3\)
\(=\left(2x^2\right)^3+3\cdot\left(2x^2\right)^2\cdot\frac13+3\cdot2x^2\cdot\left(\frac13\right)^2+\left(\frac13\right)^3\)
\(=8x^6+4x^4+\frac23x^2+\frac{1}{27}\)
b: \(\left(2x^2y-3xy\right)^3\)
\(=\left(2x^2y\right)^3-3\cdot\left(2x^2y\right)^2\cdot3xy+3\cdot2x^2y\cdot\left(3xy\right)^2-\left(3xy\right)^3\)
\(=8x^6y^3-9xy\cdot4x^4y^2+6x^2y\cdot9x^2y^2-27x^3y^3\)
\(=8x^6y^3-36x^5y^3+54x^4y^3-27x^3y^3\)
c: \(\left(-3xy^4+\frac12x^2y^2\right)^3\)
\(=\left(-3xy^4\right)^3+3\cdot\left(-3xy^4\right)^2\cdot\frac12x^2y^2+3\cdot\left(-3xy^4\right)\cdot\left(\frac12x^2y^2\right)^2+\left(\frac12x^2y^2\right)^3\)
\(=-27x^3y^{12}+\frac32x^2y^2\cdot9x^2y^8-9xy^4\cdot\frac14x^4y^4+\frac18x^6y^6\)
\(=-27x^3y^{12}+\frac{27}{2}x^4y^{10}-\frac94x^5y^8+\frac18x^6y^6\)
d: \(\left(-\frac13ab^2-2a^3b\right)^3\)
\(=-\left\lbrack\left(\frac13ab^2\right)^3+3\cdot\left(\frac13ab^2\right)^2\cdot2a^3b+3\cdot\frac13ab^2\cdot\left(2a^3b\right)^2+\left(2a^3b\right)^3\right\rbrack\)
\(=-\frac{1}{27}a^3b^6-6a^3b\cdot\frac19a^2b^4-ab^2\cdot4a^6b^2-8a^9b^3\)
\(=-\frac{1}{27}a^3b^6-\frac23a^5b^5-4a^7b^4-8a^9b^3\)
e: \(\left(x+1\right)^3-\left(x-1\right)^3-6\left(x-1\right)\left(x+1\right)\)
\(=x^3+3x^2+3x+1-\left(x^3-3x^2+3x-1\right)-6\left(x^2-1\right)\)
\(=x^3+3x^2+3x+1-x^3+3x^2-3x+1-6x^2+6\)
=8
f: x(x-1)(x+1)-(x+1)\(\left(x^2-x+1\right)\)
\(=x\left(x^2-1\right)-\left(x^3+1\right)\)
\(=x_{}^3-x-x^3-1=-x-1\)
g: \(\left(x-1\right)^3-\left(x+2\right)\left(x^2-2x+4\right)+3\left(x-4\right)\left(x+4\right)\)
\(=x^3-3x^2+3x-1-\left(x^3+8\right)+3\cdot\left(x^2-16\right)\)
\(=x^3-3x^2+3x-1-x^3-8+3x^2-48=3x-57\)
h: \(3x^2\left(x+1\right)\left(x-1\right)+\left(x^2-1\right)^3-\left(x^2-1\right)\left(x^4+x^2+1\right)\)
\(=3x^2\left(x^2-1\right)+x^6-3x^4+3x^2-1-x^6+1\)
\(=3x^4-3x^2-3x^4+3x^2=0\)
k: \(\left(x^4-3x^2+9\right)\left(x^2+3\right)+\left(3-x^2\right)^3-9x^2\left(x^2-3\right)\)
\(=x^6+27+27-27x^2+_{}9x^4-x^6-9x^4+27x^2=54\)
l: \(\left(4x+6y\right)\left(4x^2-6xy+9y^2\right)-54y^3\)
\(=2\left(2x+3y\right)\left(4x^2-6xy+9y^2\right)-54y^3\)
\(=2\cdot\left(8x^3+27y^3\right)-54y^3=16x^3+54y^3-54y^3=16x^3\)
