Solve left(x^3-2xright)^3-2(x^3-2x)

$${ \left( { x }^{ 3 } -2x \right) }^{ 3 } -2( { x }^{ 3 } -2x)$$

Evaluate

$$x\left(x^{2}-2\right)\left(\left(x\left(x^{2}-2\right)\right)^{2}-2\right)$$

Solution Steps

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(x^{3}-2x\right)^{3}.

$$\left(x^{3}\right)^{3}-6\left(x^{3}\right)^{2}x+12x^{3}x^{2}-8x^{3}-2\left(x^{3}-2x\right)$$

To raise a power to another power, multiply the exponents. Multiply 3 and 3 to get 9.

$$x^{9}-6\left(x^{3}\right)^{2}x+12x^{3}x^{2}-8x^{3}-2\left(x^{3}-2x\right)$$

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$x^{9}-6x^{6}x+12x^{3}x^{2}-8x^{3}-2\left(x^{3}-2x\right)$$

To multiply powers of the same base, add their exponents. Add 6 and 1 to get 7.

$$x^{9}-6x^{7}+12x^{3}x^{2}-8x^{3}-2\left(x^{3}-2x\right)$$

To multiply powers of the same base, add their exponents. Add 3 and 2 to get 5.

$$x^{9}-6x^{7}+12x^{5}-8x^{3}-2\left(x^{3}-2x\right)$$

Use the distributive property to multiply -2 by x^{3}-2x.

$$x^{9}-6x^{7}+12x^{5}-8x^{3}-2x^{3}+4x$$

Combine -8x^{3} and -2x^{3} to get -10x^{3}.

$$x^{9}-6x^{7}+12x^{5}-10x^{3}+4x$$

Expand

$$x^{9}-6x^{7}+12x^{5}-10x^{3}+4x$$

Solution Steps

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(x^{3}-2x\right)^{3}.

$$\left(x^{3}\right)^{3}-6\left(x^{3}\right)^{2}x+12x^{3}x^{2}-8x^{3}-2\left(x^{3}-2x\right)$$

To raise a power to another power, multiply the exponents. Multiply 3 and 3 to get 9.

$$x^{9}-6\left(x^{3}\right)^{2}x+12x^{3}x^{2}-8x^{3}-2\left(x^{3}-2x\right)$$

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$x^{9}-6x^{6}x+12x^{3}x^{2}-8x^{3}-2\left(x^{3}-2x\right)$$

To multiply powers of the same base, add their exponents. Add 6 and 1 to get 7.

$$x^{9}-6x^{7}+12x^{3}x^{2}-8x^{3}-2\left(x^{3}-2x\right)$$

To multiply powers of the same base, add their exponents. Add 3 and 2 to get 5.

$$x^{9}-6x^{7}+12x^{5}-8x^{3}-2\left(x^{3}-2x\right)$$

Use the distributive property to multiply -2 by x^{3}-2x.

$$x^{9}-6x^{7}+12x^{5}-8x^{3}-2x^{3}+4x$$

Combine -8x^{3} and -2x^{3} to get -10x^{3}.

$$x^{9}-6x^{7}+12x^{5}-10x^{3}+4x$$