Solve (8-12x^{2})^{3}*(-6x)*(x^{3}+3)^{3}+(3x^{2}+9)^2(7x^6)*(2-3x^2)^3

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

Evaluate

$$3x\left(2-3x^{2}\right)^{3}\left(-107x^{9}+126x^{7}-1152x^{6}+189x^{5}-3456x^{3}-3456\right)$$

Solution Steps

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(8-12x^{2}\right)^{3}.

$$\left(512-2304x^{2}+3456\left(x^{2}\right)^{2}-1728\left(x^{2}\right)^{3}\right)\left(-6\right)x\left(x^{3}+3\right)^{3}+\left(3x^{2}+9\right)^{2}\times 7x^{6}\left(2-3x^{2}\right)^{3}$$

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

$$\left(512-2304x^{2}+3456x^{4}-1728\left(x^{2}\right)^{3}\right)\left(-6\right)x\left(x^{3}+3\right)^{3}+\left(3x^{2}+9\right)^{2}\times 7x^{6}\left(2-3x^{2}\right)^{3}$$

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

$$\left(512-2304x^{2}+3456x^{4}-1728x^{6}\right)\left(-6\right)x\left(x^{3}+3\right)^{3}+\left(3x^{2}+9\right)^{2}\times 7x^{6}\left(2-3x^{2}\right)^{3}$$

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

$$\left(512-2304x^{2}+3456x^{4}-1728x^{6}\right)\left(-6\right)x\left(\left(x^{3}\right)^{3}+9\left(x^{3}\right)^{2}+27x^{3}+27\right)+\left(3x^{2}+9\right)^{2}\times 7x^{6}\left(2-3x^{2}\right)^{3}$$

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

$$\left(512-2304x^{2}+3456x^{4}-1728x^{6}\right)\left(-6\right)x\left(x^{9}+9\left(x^{3}\right)^{2}+27x^{3}+27\right)+\left(3x^{2}+9\right)^{2}\times 7x^{6}\left(2-3x^{2}\right)^{3}$$

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

$$\left(512-2304x^{2}+3456x^{4}-1728x^{6}\right)\left(-6\right)x\left(x^{9}+9x^{6}+27x^{3}+27\right)+\left(3x^{2}+9\right)^{2}\times 7x^{6}\left(2-3x^{2}\right)^{3}$$

Use the distributive property to multiply 512-2304x^{2}+3456x^{4}-1728x^{6} by -6.

$$\left(-3072+13824x^{2}-20736x^{4}+10368x^{6}\right)x\left(x^{9}+9x^{6}+27x^{3}+27\right)+\left(3x^{2}+9\right)^{2}\times 7x^{6}\left(2-3x^{2}\right)^{3}$$

Use the distributive property to multiply -3072+13824x^{2}-20736x^{4}+10368x^{6} by x.

$$\left(-3072x+13824x^{3}-20736x^{5}+10368x^{7}\right)\left(x^{9}+9x^{6}+27x^{3}+27\right)+\left(3x^{2}+9\right)^{2}\times 7x^{6}\left(2-3x^{2}\right)^{3}$$

Use the distributive property to multiply -3072x+13824x^{3}-20736x^{5}+10368x^{7} by x^{9}+9x^{6}+27x^{3}+27 and combine like terms.

$$276864x^{10}+252288x^{7}-82944x^{4}-82944x+13824x^{12}+124416x^{9}+373248x^{6}+373248x^{3}-20736x^{14}-186624x^{11}-559872x^{8}-559872x^{5}+10368x^{16}+93312x^{13}+\left(3x^{2}+9\right)^{2}\times 7x^{6}\left(2-3x^{2}\right)^{3}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3x^{2}+9\right)^{2}.

$$276864x^{10}+252288x^{7}-82944x^{4}-82944x+13824x^{12}+124416x^{9}+373248x^{6}+373248x^{3}-20736x^{14}-186624x^{11}-559872x^{8}-559872x^{5}+10368x^{16}+93312x^{13}+\left(9\left(x^{2}\right)^{2}+54x^{2}+81\right)\times 7x^{6}\left(2-3x^{2}\right)^{3}$$

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

$$276864x^{10}+252288x^{7}-82944x^{4}-82944x+13824x^{12}+124416x^{9}+373248x^{6}+373248x^{3}-20736x^{14}-186624x^{11}-559872x^{8}-559872x^{5}+10368x^{16}+93312x^{13}+\left(9x^{4}+54x^{2}+81\right)\times 7x^{6}\left(2-3x^{2}\right)^{3}$$

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

$$276864x^{10}+252288x^{7}-82944x^{4}-82944x+13824x^{12}+124416x^{9}+373248x^{6}+373248x^{3}-20736x^{14}-186624x^{11}-559872x^{8}-559872x^{5}+10368x^{16}+93312x^{13}+\left(9x^{4}+54x^{2}+81\right)\times 7x^{6}\left(8-36x^{2}+54\left(x^{2}\right)^{2}-27\left(x^{2}\right)^{3}\right)$$

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

$$276864x^{10}+252288x^{7}-82944x^{4}-82944x+13824x^{12}+124416x^{9}+373248x^{6}+373248x^{3}-20736x^{14}-186624x^{11}-559872x^{8}-559872x^{5}+10368x^{16}+93312x^{13}+\left(9x^{4}+54x^{2}+81\right)\times 7x^{6}\left(8-36x^{2}+54x^{4}-27\left(x^{2}\right)^{3}\right)$$

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

$$276864x^{10}+252288x^{7}-82944x^{4}-82944x+13824x^{12}+124416x^{9}+373248x^{6}+373248x^{3}-20736x^{14}-186624x^{11}-559872x^{8}-559872x^{5}+10368x^{16}+93312x^{13}+\left(9x^{4}+54x^{2}+81\right)\times 7x^{6}\left(8-36x^{2}+54x^{4}-27x^{6}\right)$$

Use the distributive property to multiply 9x^{4}+54x^{2}+81 by 7.

$$276864x^{10}+252288x^{7}-82944x^{4}-82944x+13824x^{12}+124416x^{9}+373248x^{6}+373248x^{3}-20736x^{14}-186624x^{11}-559872x^{8}-559872x^{5}+10368x^{16}+93312x^{13}+\left(63x^{4}+378x^{2}+567\right)x^{6}\left(8-36x^{2}+54x^{4}-27x^{6}\right)$$

Use the distributive property to multiply 63x^{4}+378x^{2}+567 by x^{6}.

$$276864x^{10}+252288x^{7}-82944x^{4}-82944x+13824x^{12}+124416x^{9}+373248x^{6}+373248x^{3}-20736x^{14}-186624x^{11}-559872x^{8}-559872x^{5}+10368x^{16}+93312x^{13}+\left(63x^{10}+378x^{8}+567x^{6}\right)\left(8-36x^{2}+54x^{4}-27x^{6}\right)$$

Use the distributive property to multiply 63x^{10}+378x^{8}+567x^{6} by 8-36x^{2}+54x^{4}-27x^{6} and combine like terms.

$$276864x^{10}+252288x^{7}-82944x^{4}-82944x+13824x^{12}+124416x^{9}+373248x^{6}+373248x^{3}-20736x^{14}-186624x^{11}-559872x^{8}-559872x^{5}+10368x^{16}+93312x^{13}+17514x^{10}+2835x^{12}-6804x^{14}-1701x^{16}-17388x^{8}+4536x^{6}$$

Combine 276864x^{10} and 17514x^{10} to get 294378x^{10}.

$$294378x^{10}+252288x^{7}-82944x^{4}-82944x+13824x^{12}+124416x^{9}+373248x^{6}+373248x^{3}-20736x^{14}-186624x^{11}-559872x^{8}-559872x^{5}+10368x^{16}+93312x^{13}+2835x^{12}-6804x^{14}-1701x^{16}-17388x^{8}+4536x^{6}$$

Combine 13824x^{12} and 2835x^{12} to get 16659x^{12}.

$$294378x^{10}+252288x^{7}-82944x^{4}-82944x+16659x^{12}+124416x^{9}+373248x^{6}+373248x^{3}-20736x^{14}-186624x^{11}-559872x^{8}-559872x^{5}+10368x^{16}+93312x^{13}-6804x^{14}-1701x^{16}-17388x^{8}+4536x^{6}$$

Combine -20736x^{14} and -6804x^{14} to get -27540x^{14}.

$$294378x^{10}+252288x^{7}-82944x^{4}-82944x+16659x^{12}+124416x^{9}+373248x^{6}+373248x^{3}-27540x^{14}-186624x^{11}-559872x^{8}-559872x^{5}+10368x^{16}+93312x^{13}-1701x^{16}-17388x^{8}+4536x^{6}$$

Combine 10368x^{16} and -1701x^{16} to get 8667x^{16}.

$$294378x^{10}+252288x^{7}-82944x^{4}-82944x+16659x^{12}+124416x^{9}+373248x^{6}+373248x^{3}-27540x^{14}-186624x^{11}-559872x^{8}-559872x^{5}+8667x^{16}+93312x^{13}-17388x^{8}+4536x^{6}$$

Combine -559872x^{8} and -17388x^{8} to get -577260x^{8}.

$$294378x^{10}+252288x^{7}-82944x^{4}-82944x+16659x^{12}+124416x^{9}+373248x^{6}+373248x^{3}-27540x^{14}-186624x^{11}-577260x^{8}-559872x^{5}+8667x^{16}+93312x^{13}+4536x^{6}$$

Combine 373248x^{6} and 4536x^{6} to get 377784x^{6}.

$$294378x^{10}+252288x^{7}-82944x^{4}-82944x+16659x^{12}+124416x^{9}+377784x^{6}+373248x^{3}-27540x^{14}-186624x^{11}-577260x^{8}-559872x^{5}+8667x^{16}+93312x^{13}$$

Expand

$$8667x^{16}-27540x^{14}+93312x^{13}+16659x^{12}-186624x^{11}+294378x^{10}+124416x^{9}-577260x^{8}+252288x^{7}+377784x^{6}-559872x^{5}-82944x^{4}+373248x^{3}-82944x$$

Solution Steps

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(8-12x^{2}\right)^{3}.

$$\left(512-2304x^{2}+3456\left(x^{2}\right)^{2}-1728\left(x^{2}\right)^{3}\right)\left(-6\right)x\left(x^{3}+3\right)^{3}+\left(3x^{2}+9\right)^{2}\times 7x^{6}\left(2-3x^{2}\right)^{3}$$

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

$$\left(512-2304x^{2}+3456x^{4}-1728\left(x^{2}\right)^{3}\right)\left(-6\right)x\left(x^{3}+3\right)^{3}+\left(3x^{2}+9\right)^{2}\times 7x^{6}\left(2-3x^{2}\right)^{3}$$

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

$$\left(512-2304x^{2}+3456x^{4}-1728x^{6}\right)\left(-6\right)x\left(x^{3}+3\right)^{3}+\left(3x^{2}+9\right)^{2}\times 7x^{6}\left(2-3x^{2}\right)^{3}$$

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

$$\left(512-2304x^{2}+3456x^{4}-1728x^{6}\right)\left(-6\right)x\left(\left(x^{3}\right)^{3}+9\left(x^{3}\right)^{2}+27x^{3}+27\right)+\left(3x^{2}+9\right)^{2}\times 7x^{6}\left(2-3x^{2}\right)^{3}$$

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

$$\left(512-2304x^{2}+3456x^{4}-1728x^{6}\right)\left(-6\right)x\left(x^{9}+9\left(x^{3}\right)^{2}+27x^{3}+27\right)+\left(3x^{2}+9\right)^{2}\times 7x^{6}\left(2-3x^{2}\right)^{3}$$

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

$$\left(512-2304x^{2}+3456x^{4}-1728x^{6}\right)\left(-6\right)x\left(x^{9}+9x^{6}+27x^{3}+27\right)+\left(3x^{2}+9\right)^{2}\times 7x^{6}\left(2-3x^{2}\right)^{3}$$

Use the distributive property to multiply 512-2304x^{2}+3456x^{4}-1728x^{6} by -6.

$$\left(-3072+13824x^{2}-20736x^{4}+10368x^{6}\right)x\left(x^{9}+9x^{6}+27x^{3}+27\right)+\left(3x^{2}+9\right)^{2}\times 7x^{6}\left(2-3x^{2}\right)^{3}$$

Use the distributive property to multiply -3072+13824x^{2}-20736x^{4}+10368x^{6} by x.

$$\left(-3072x+13824x^{3}-20736x^{5}+10368x^{7}\right)\left(x^{9}+9x^{6}+27x^{3}+27\right)+\left(3x^{2}+9\right)^{2}\times 7x^{6}\left(2-3x^{2}\right)^{3}$$

Use the distributive property to multiply -3072x+13824x^{3}-20736x^{5}+10368x^{7} by x^{9}+9x^{6}+27x^{3}+27 and combine like terms.

$$276864x^{10}+252288x^{7}-82944x^{4}-82944x+13824x^{12}+124416x^{9}+373248x^{6}+373248x^{3}-20736x^{14}-186624x^{11}-559872x^{8}-559872x^{5}+10368x^{16}+93312x^{13}+\left(3x^{2}+9\right)^{2}\times 7x^{6}\left(2-3x^{2}\right)^{3}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3x^{2}+9\right)^{2}.

$$276864x^{10}+252288x^{7}-82944x^{4}-82944x+13824x^{12}+124416x^{9}+373248x^{6}+373248x^{3}-20736x^{14}-186624x^{11}-559872x^{8}-559872x^{5}+10368x^{16}+93312x^{13}+\left(9\left(x^{2}\right)^{2}+54x^{2}+81\right)\times 7x^{6}\left(2-3x^{2}\right)^{3}$$

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

$$276864x^{10}+252288x^{7}-82944x^{4}-82944x+13824x^{12}+124416x^{9}+373248x^{6}+373248x^{3}-20736x^{14}-186624x^{11}-559872x^{8}-559872x^{5}+10368x^{16}+93312x^{13}+\left(9x^{4}+54x^{2}+81\right)\times 7x^{6}\left(2-3x^{2}\right)^{3}$$

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

$$276864x^{10}+252288x^{7}-82944x^{4}-82944x+13824x^{12}+124416x^{9}+373248x^{6}+373248x^{3}-20736x^{14}-186624x^{11}-559872x^{8}-559872x^{5}+10368x^{16}+93312x^{13}+\left(9x^{4}+54x^{2}+81\right)\times 7x^{6}\left(8-36x^{2}+54\left(x^{2}\right)^{2}-27\left(x^{2}\right)^{3}\right)$$

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

$$276864x^{10}+252288x^{7}-82944x^{4}-82944x+13824x^{12}+124416x^{9}+373248x^{6}+373248x^{3}-20736x^{14}-186624x^{11}-559872x^{8}-559872x^{5}+10368x^{16}+93312x^{13}+\left(9x^{4}+54x^{2}+81\right)\times 7x^{6}\left(8-36x^{2}+54x^{4}-27\left(x^{2}\right)^{3}\right)$$

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

$$276864x^{10}+252288x^{7}-82944x^{4}-82944x+13824x^{12}+124416x^{9}+373248x^{6}+373248x^{3}-20736x^{14}-186624x^{11}-559872x^{8}-559872x^{5}+10368x^{16}+93312x^{13}+\left(9x^{4}+54x^{2}+81\right)\times 7x^{6}\left(8-36x^{2}+54x^{4}-27x^{6}\right)$$

Use the distributive property to multiply 9x^{4}+54x^{2}+81 by 7.

$$276864x^{10}+252288x^{7}-82944x^{4}-82944x+13824x^{12}+124416x^{9}+373248x^{6}+373248x^{3}-20736x^{14}-186624x^{11}-559872x^{8}-559872x^{5}+10368x^{16}+93312x^{13}+\left(63x^{4}+378x^{2}+567\right)x^{6}\left(8-36x^{2}+54x^{4}-27x^{6}\right)$$

Use the distributive property to multiply 63x^{4}+378x^{2}+567 by x^{6}.

$$276864x^{10}+252288x^{7}-82944x^{4}-82944x+13824x^{12}+124416x^{9}+373248x^{6}+373248x^{3}-20736x^{14}-186624x^{11}-559872x^{8}-559872x^{5}+10368x^{16}+93312x^{13}+\left(63x^{10}+378x^{8}+567x^{6}\right)\left(8-36x^{2}+54x^{4}-27x^{6}\right)$$

Use the distributive property to multiply 63x^{10}+378x^{8}+567x^{6} by 8-36x^{2}+54x^{4}-27x^{6} and combine like terms.

$$276864x^{10}+252288x^{7}-82944x^{4}-82944x+13824x^{12}+124416x^{9}+373248x^{6}+373248x^{3}-20736x^{14}-186624x^{11}-559872x^{8}-559872x^{5}+10368x^{16}+93312x^{13}+17514x^{10}+2835x^{12}-6804x^{14}-1701x^{16}-17388x^{8}+4536x^{6}$$

Combine 276864x^{10} and 17514x^{10} to get 294378x^{10}.

$$294378x^{10}+252288x^{7}-82944x^{4}-82944x+13824x^{12}+124416x^{9}+373248x^{6}+373248x^{3}-20736x^{14}-186624x^{11}-559872x^{8}-559872x^{5}+10368x^{16}+93312x^{13}+2835x^{12}-6804x^{14}-1701x^{16}-17388x^{8}+4536x^{6}$$

Combine 13824x^{12} and 2835x^{12} to get 16659x^{12}.

$$294378x^{10}+252288x^{7}-82944x^{4}-82944x+16659x^{12}+124416x^{9}+373248x^{6}+373248x^{3}-20736x^{14}-186624x^{11}-559872x^{8}-559872x^{5}+10368x^{16}+93312x^{13}-6804x^{14}-1701x^{16}-17388x^{8}+4536x^{6}$$

Combine -20736x^{14} and -6804x^{14} to get -27540x^{14}.

$$294378x^{10}+252288x^{7}-82944x^{4}-82944x+16659x^{12}+124416x^{9}+373248x^{6}+373248x^{3}-27540x^{14}-186624x^{11}-559872x^{8}-559872x^{5}+10368x^{16}+93312x^{13}-1701x^{16}-17388x^{8}+4536x^{6}$$

Combine 10368x^{16} and -1701x^{16} to get 8667x^{16}.

$$294378x^{10}+252288x^{7}-82944x^{4}-82944x+16659x^{12}+124416x^{9}+373248x^{6}+373248x^{3}-27540x^{14}-186624x^{11}-559872x^{8}-559872x^{5}+8667x^{16}+93312x^{13}-17388x^{8}+4536x^{6}$$

Combine -559872x^{8} and -17388x^{8} to get -577260x^{8}.

$$294378x^{10}+252288x^{7}-82944x^{4}-82944x+16659x^{12}+124416x^{9}+373248x^{6}+373248x^{3}-27540x^{14}-186624x^{11}-577260x^{8}-559872x^{5}+8667x^{16}+93312x^{13}+4536x^{6}$$

Combine 373248x^{6} and 4536x^{6} to get 377784x^{6}.

$$294378x^{10}+252288x^{7}-82944x^{4}-82944x+16659x^{12}+124416x^{9}+377784x^{6}+373248x^{3}-27540x^{14}-186624x^{11}-577260x^{8}-559872x^{5}+8667x^{16}+93312x^{13}$$