Solve (3x)^{4}(1/3ax)^{2}-[(2x-4x)^{2}(a^{2}x^{4}+1/2a^{2}x^{4})+(7a^2x^6-6a^2x^6)]

$$( 3 x ) ^ { 4 } ( \frac { 1 } { 3 } a x ) ^ { 2 } - [ ( 2 x - 4 x ) ^ { 2 } ( a ^ { 2 } x ^ { 4 } + \frac { 1 } { 2 } a ^ { 2 } x ^ { 4 } ) + ( 7 a ^ { 2 } x ^ { 6 } - 6 a ^ { 2 } x ^ { 6 } ) ]$$

Evaluate

$$2a^{2}x^{6}$$

Solution Steps

Expand \left(3x\right)^{4}.

$$3^{4}x^{4}\times \left(\frac{1}{3}ax\right)^{2}-\left(\left(2x-4x\right)^{2}\left(a^{2}x^{4}+\frac{1}{2}a^{2}x^{4}\right)+7a^{2}x^{6}-6a^{2}x^{6}\right)$$

Calculate 3 to the power of 4 and get 81.

$$81x^{4}\times \left(\frac{1}{3}ax\right)^{2}-\left(\left(2x-4x\right)^{2}\left(a^{2}x^{4}+\frac{1}{2}a^{2}x^{4}\right)+7a^{2}x^{6}-6a^{2}x^{6}\right)$$

Expand \left(\frac{1}{3}ax\right)^{2}.

$$81x^{4}\times \left(\frac{1}{3}\right)^{2}a^{2}x^{2}-\left(\left(2x-4x\right)^{2}\left(a^{2}x^{4}+\frac{1}{2}a^{2}x^{4}\right)+7a^{2}x^{6}-6a^{2}x^{6}\right)$$

Calculate \frac{1}{3} to the power of 2 and get \frac{1}{9}.

$$81x^{4}\times \frac{1}{9}a^{2}x^{2}-\left(\left(2x-4x\right)^{2}\left(a^{2}x^{4}+\frac{1}{2}a^{2}x^{4}\right)+7a^{2}x^{6}-6a^{2}x^{6}\right)$$

Multiply 81 and \frac{1}{9} to get 9.

$$9x^{4}a^{2}x^{2}-\left(\left(2x-4x\right)^{2}\left(a^{2}x^{4}+\frac{1}{2}a^{2}x^{4}\right)+7a^{2}x^{6}-6a^{2}x^{6}\right)$$

To multiply powers of the same base, add their exponents. Add 4 and 2 to get 6.

$$9x^{6}a^{2}-\left(\left(2x-4x\right)^{2}\left(a^{2}x^{4}+\frac{1}{2}a^{2}x^{4}\right)+7a^{2}x^{6}-6a^{2}x^{6}\right)$$

Combine 2x and -4x to get -2x.

$$9x^{6}a^{2}-\left(\left(-2x\right)^{2}\left(a^{2}x^{4}+\frac{1}{2}a^{2}x^{4}\right)+7a^{2}x^{6}-6a^{2}x^{6}\right)$$

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

$$9x^{6}a^{2}-\left(\left(-2\right)^{2}x^{2}\left(a^{2}x^{4}+\frac{1}{2}a^{2}x^{4}\right)+7a^{2}x^{6}-6a^{2}x^{6}\right)$$

Calculate -2 to the power of 2 and get 4.

$$9x^{6}a^{2}-\left(4x^{2}\left(a^{2}x^{4}+\frac{1}{2}a^{2}x^{4}\right)+7a^{2}x^{6}-6a^{2}x^{6}\right)$$

Combine a^{2}x^{4} and \frac{1}{2}a^{2}x^{4} to get \frac{3}{2}a^{2}x^{4}.

$$9x^{6}a^{2}-\left(4x^{2}\times \frac{3}{2}a^{2}x^{4}+7a^{2}x^{6}-6a^{2}x^{6}\right)$$

Multiply 4 and \frac{3}{2} to get 6.

$$9x^{6}a^{2}-\left(6x^{2}a^{2}x^{4}+7a^{2}x^{6}-6a^{2}x^{6}\right)$$

To multiply powers of the same base, add their exponents. Add 2 and 4 to get 6.

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

Combine 7a^{2}x^{6} and -6a^{2}x^{6} to get a^{2}x^{6}.

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

Combine 6x^{6}a^{2} and a^{2}x^{6} to get 7x^{6}a^{2}.

$$9x^{6}a^{2}-7x^{6}a^{2}$$

Combine 9x^{6}a^{2} and -7x^{6}a^{2} to get 2x^{6}a^{2}.

$$2x^{6}a^{2}$$

Expand

$$2a^{2}x^{6}$$

Solution Steps

Expand \left(3x\right)^{4}.

$$3^{4}x^{4}\times \left(\frac{1}{3}ax\right)^{2}-\left(\left(2x-4x\right)^{2}\left(a^{2}x^{4}+\frac{1}{2}a^{2}x^{4}\right)+7a^{2}x^{6}-6a^{2}x^{6}\right)$$

Calculate 3 to the power of 4 and get 81.

$$81x^{4}\times \left(\frac{1}{3}ax\right)^{2}-\left(\left(2x-4x\right)^{2}\left(a^{2}x^{4}+\frac{1}{2}a^{2}x^{4}\right)+7a^{2}x^{6}-6a^{2}x^{6}\right)$$

Expand \left(\frac{1}{3}ax\right)^{2}.

$$81x^{4}\times \left(\frac{1}{3}\right)^{2}a^{2}x^{2}-\left(\left(2x-4x\right)^{2}\left(a^{2}x^{4}+\frac{1}{2}a^{2}x^{4}\right)+7a^{2}x^{6}-6a^{2}x^{6}\right)$$

Calculate \frac{1}{3} to the power of 2 and get \frac{1}{9}.

$$81x^{4}\times \frac{1}{9}a^{2}x^{2}-\left(\left(2x-4x\right)^{2}\left(a^{2}x^{4}+\frac{1}{2}a^{2}x^{4}\right)+7a^{2}x^{6}-6a^{2}x^{6}\right)$$

Multiply 81 and \frac{1}{9} to get 9.

$$9x^{4}a^{2}x^{2}-\left(\left(2x-4x\right)^{2}\left(a^{2}x^{4}+\frac{1}{2}a^{2}x^{4}\right)+7a^{2}x^{6}-6a^{2}x^{6}\right)$$

To multiply powers of the same base, add their exponents. Add 4 and 2 to get 6.

$$9x^{6}a^{2}-\left(\left(2x-4x\right)^{2}\left(a^{2}x^{4}+\frac{1}{2}a^{2}x^{4}\right)+7a^{2}x^{6}-6a^{2}x^{6}\right)$$

Combine 2x and -4x to get -2x.

$$9x^{6}a^{2}-\left(\left(-2x\right)^{2}\left(a^{2}x^{4}+\frac{1}{2}a^{2}x^{4}\right)+7a^{2}x^{6}-6a^{2}x^{6}\right)$$

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

$$9x^{6}a^{2}-\left(\left(-2\right)^{2}x^{2}\left(a^{2}x^{4}+\frac{1}{2}a^{2}x^{4}\right)+7a^{2}x^{6}-6a^{2}x^{6}\right)$$

Calculate -2 to the power of 2 and get 4.

$$9x^{6}a^{2}-\left(4x^{2}\left(a^{2}x^{4}+\frac{1}{2}a^{2}x^{4}\right)+7a^{2}x^{6}-6a^{2}x^{6}\right)$$

Combine a^{2}x^{4} and \frac{1}{2}a^{2}x^{4} to get \frac{3}{2}a^{2}x^{4}.

$$9x^{6}a^{2}-\left(4x^{2}\times \frac{3}{2}a^{2}x^{4}+7a^{2}x^{6}-6a^{2}x^{6}\right)$$

Multiply 4 and \frac{3}{2} to get 6.

$$9x^{6}a^{2}-\left(6x^{2}a^{2}x^{4}+7a^{2}x^{6}-6a^{2}x^{6}\right)$$

To multiply powers of the same base, add their exponents. Add 2 and 4 to get 6.

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

Combine 7a^{2}x^{6} and -6a^{2}x^{6} to get a^{2}x^{6}.

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

Combine 6x^{6}a^{2} and a^{2}x^{6} to get 7x^{6}a^{2}.

$$9x^{6}a^{2}-7x^{6}a^{2}$$

Combine 9x^{6}a^{2} and -7x^{6}a^{2} to get 2x^{6}a^{2}.

$$2x^{6}a^{2}$$