Solve {l}{(-3x^{3})^{2}-[(2x)^{2}]^3}{-9x^6(1+x^2)^3}

$$\left. \begin{array} { l } { ( - 3 x ^ { 3 } ) ^ { 2 } - [ ( 2 x ) ^ { 2 } ] ^ { 3 } } \\ { - 9 x ^ { 6 } ( 1 + x ^ { 2 } ) ^ { 3 } } \end{array} \right.$$

Evaluate

$$-9x^{12}-27x^{10}-27x^{8}-64x^{6}$$

Solution Steps

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

$$\left(-3x^{3}\right)^{2}-\left(2x\right)^{6}-9x^{6}\left(1+x^{2}\right)^{3}$$

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

$$\left(-3\right)^{2}\left(x^{3}\right)^{2}-\left(2x\right)^{6}-9x^{6}\left(1+x^{2}\right)^{3}$$

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

$$\left(-3\right)^{2}x^{6}-\left(2x\right)^{6}-9x^{6}\left(1+x^{2}\right)^{3}$$

Calculate -3 to the power of 2 and get 9.

$$9x^{6}-\left(2x\right)^{6}-9x^{6}\left(1+x^{2}\right)^{3}$$

Expand \left(2x\right)^{6}.

$$9x^{6}-2^{6}x^{6}-9x^{6}\left(1+x^{2}\right)^{3}$$

Calculate 2 to the power of 6 and get 64.

$$9x^{6}-64x^{6}-9x^{6}\left(1+x^{2}\right)^{3}$$

Combine 9x^{6} and -64x^{6} to get -55x^{6}.

$$-55x^{6}-9x^{6}\left(1+x^{2}\right)^{3}$$

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(1+x^{2}\right)^{3}.

$$-55x^{6}-9x^{6}\left(1+3x^{2}+3\left(x^{2}\right)^{2}+\left(x^{2}\right)^{3}\right)$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$-55x^{6}-9x^{6}\left(1+3x^{2}+3x^{4}+\left(x^{2}\right)^{3}\right)$$

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

$$-55x^{6}-9x^{6}\left(1+3x^{2}+3x^{4}+x^{6}\right)$$

Use the distributive property to multiply -9x^{6} by 1+3x^{2}+3x^{4}+x^{6}.

$$-55x^{6}-9x^{6}-27x^{8}-27x^{10}-9x^{12}$$

Combine -55x^{6} and -9x^{6} to get -64x^{6}.

$$-64x^{6}-27x^{8}-27x^{10}-9x^{12}$$

Expand

$$-9x^{12}-27x^{10}-27x^{8}-64x^{6}$$

Solution Steps

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

$$\left(-3x^{3}\right)^{2}-\left(2x\right)^{6}-9x^{6}\left(1+x^{2}\right)^{3}$$

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

$$\left(-3\right)^{2}\left(x^{3}\right)^{2}-\left(2x\right)^{6}-9x^{6}\left(1+x^{2}\right)^{3}$$

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

$$\left(-3\right)^{2}x^{6}-\left(2x\right)^{6}-9x^{6}\left(1+x^{2}\right)^{3}$$

Calculate -3 to the power of 2 and get 9.

$$9x^{6}-\left(2x\right)^{6}-9x^{6}\left(1+x^{2}\right)^{3}$$

Expand \left(2x\right)^{6}.

$$9x^{6}-2^{6}x^{6}-9x^{6}\left(1+x^{2}\right)^{3}$$

Calculate 2 to the power of 6 and get 64.

$$9x^{6}-64x^{6}-9x^{6}\left(1+x^{2}\right)^{3}$$

Combine 9x^{6} and -64x^{6} to get -55x^{6}.

$$-55x^{6}-9x^{6}\left(1+x^{2}\right)^{3}$$

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(1+x^{2}\right)^{3}.

$$-55x^{6}-9x^{6}\left(1+3x^{2}+3\left(x^{2}\right)^{2}+\left(x^{2}\right)^{3}\right)$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$-55x^{6}-9x^{6}\left(1+3x^{2}+3x^{4}+\left(x^{2}\right)^{3}\right)$$

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

$$-55x^{6}-9x^{6}\left(1+3x^{2}+3x^{4}+x^{6}\right)$$

Use the distributive property to multiply -9x^{6} by 1+3x^{2}+3x^{4}+x^{6}.

$$-55x^{6}-9x^{6}-27x^{8}-27x^{10}-9x^{12}$$

Combine -55x^{6} and -9x^{6} to get -64x^{6}.

$$-64x^{6}-27x^{8}-27x^{10}-9x^{12}$$