Solve [(-xy^{2}z)^{3}]^{2}-(-2x^{2}y^{4}z^{2})^{3}-3*(-2x^{3}y^{4}z^{3})^2*(y^2-1/2y^2)^2

$$[ ( - x y ^ { 2 } z ) ^ { 3 } ] ^ { 2 } - ( - 2 x ^ { 2 } y ^ { 4 } z ^ { 2 } ) ^ { 3 } - 3 \cdot ( - 2 x ^ { 3 } y ^ { 4 } z ^ { 3 } ) ^ { 2 } \cdot ( y ^ { 2 } - \frac { 1 } { 2 } y ^ { 2 } ) ^ { 2 }$$

Evaluate

$$6y^{12}\left(xz\right)^{6}$$

Solution Steps

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

$$\left(\left(-x\right)y^{2}z\right)^{6}-\left(-2x^{2}y^{4}z^{2}\right)^{3}-3\left(-2x^{3}y^{4}z^{3}\right)^{2}\left(y^{2}-\frac{1}{2}y^{2}\right)^{2}$$

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

$$\left(-x\right)^{6}\left(y^{2}\right)^{6}z^{6}-\left(-2x^{2}y^{4}z^{2}\right)^{3}-3\left(-2x^{3}y^{4}z^{3}\right)^{2}\left(y^{2}-\frac{1}{2}y^{2}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 6 to get 12.

$$\left(-x\right)^{6}y^{12}z^{6}-\left(-2x^{2}y^{4}z^{2}\right)^{3}-3\left(-2x^{3}y^{4}z^{3}\right)^{2}\left(y^{2}-\frac{1}{2}y^{2}\right)^{2}$$

Expand \left(-2x^{2}y^{4}z^{2}\right)^{3}.

$$\left(-x\right)^{6}y^{12}z^{6}-\left(-2\right)^{3}\left(x^{2}\right)^{3}\left(y^{4}\right)^{3}\left(z^{2}\right)^{3}-3\left(-2x^{3}y^{4}z^{3}\right)^{2}\left(y^{2}-\frac{1}{2}y^{2}\right)^{2}$$

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

$$\left(-x\right)^{6}y^{12}z^{6}-\left(-2\right)^{3}x^{6}\left(y^{4}\right)^{3}\left(z^{2}\right)^{3}-3\left(-2x^{3}y^{4}z^{3}\right)^{2}\left(y^{2}-\frac{1}{2}y^{2}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 3 to get 12.

$$\left(-x\right)^{6}y^{12}z^{6}-\left(-2\right)^{3}x^{6}y^{12}\left(z^{2}\right)^{3}-3\left(-2x^{3}y^{4}z^{3}\right)^{2}\left(y^{2}-\frac{1}{2}y^{2}\right)^{2}$$

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

$$\left(-x\right)^{6}y^{12}z^{6}-\left(-2\right)^{3}x^{6}y^{12}z^{6}-3\left(-2x^{3}y^{4}z^{3}\right)^{2}\left(y^{2}-\frac{1}{2}y^{2}\right)^{2}$$

Calculate -2 to the power of 3 and get -8.

$$\left(-x\right)^{6}y^{12}z^{6}-\left(-8x^{6}y^{12}z^{6}\right)-3\left(-2x^{3}y^{4}z^{3}\right)^{2}\left(y^{2}-\frac{1}{2}y^{2}\right)^{2}$$

The opposite of -8x^{6}y^{12}z^{6} is 8x^{6}y^{12}z^{6}.

$$\left(-x\right)^{6}y^{12}z^{6}+8x^{6}y^{12}z^{6}-3\left(-2x^{3}y^{4}z^{3}\right)^{2}\left(y^{2}-\frac{1}{2}y^{2}\right)^{2}$$

Expand \left(-2x^{3}y^{4}z^{3}\right)^{2}.

$$\left(-x\right)^{6}y^{12}z^{6}+8x^{6}y^{12}z^{6}-3\left(-2\right)^{2}\left(x^{3}\right)^{2}\left(y^{4}\right)^{2}\left(z^{3}\right)^{2}\left(y^{2}-\frac{1}{2}y^{2}\right)^{2}$$

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

$$\left(-x\right)^{6}y^{12}z^{6}+8x^{6}y^{12}z^{6}-3\left(-2\right)^{2}x^{6}\left(y^{4}\right)^{2}\left(z^{3}\right)^{2}\left(y^{2}-\frac{1}{2}y^{2}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 2 to get 8.

$$\left(-x\right)^{6}y^{12}z^{6}+8x^{6}y^{12}z^{6}-3\left(-2\right)^{2}x^{6}y^{8}\left(z^{3}\right)^{2}\left(y^{2}-\frac{1}{2}y^{2}\right)^{2}$$

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

$$\left(-x\right)^{6}y^{12}z^{6}+8x^{6}y^{12}z^{6}-3\left(-2\right)^{2}x^{6}y^{8}z^{6}\left(y^{2}-\frac{1}{2}y^{2}\right)^{2}$$

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

$$\left(-x\right)^{6}y^{12}z^{6}+8x^{6}y^{12}z^{6}-3\times 4x^{6}y^{8}z^{6}\left(y^{2}-\frac{1}{2}y^{2}\right)^{2}$$

Multiply 3 and 4 to get 12.

$$\left(-x\right)^{6}y^{12}z^{6}+8x^{6}y^{12}z^{6}-12x^{6}y^{8}z^{6}\left(y^{2}-\frac{1}{2}y^{2}\right)^{2}$$

Combine y^{2} and -\frac{1}{2}y^{2} to get \frac{1}{2}y^{2}.

$$\left(-x\right)^{6}y^{12}z^{6}+8x^{6}y^{12}z^{6}-12x^{6}y^{8}z^{6}\times \left(\frac{1}{2}y^{2}\right)^{2}$$

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

$$\left(-x\right)^{6}y^{12}z^{6}+8x^{6}y^{12}z^{6}-12x^{6}y^{8}z^{6}\times \left(\frac{1}{2}\right)^{2}\left(y^{2}\right)^{2}$$

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

$$\left(-x\right)^{6}y^{12}z^{6}+8x^{6}y^{12}z^{6}-12x^{6}y^{8}z^{6}\times \left(\frac{1}{2}\right)^{2}y^{4}$$

Calculate \frac{1}{2} to the power of 2 and get \frac{1}{4}.

$$\left(-x\right)^{6}y^{12}z^{6}+8x^{6}y^{12}z^{6}-12x^{6}y^{8}z^{6}\times \frac{1}{4}y^{4}$$

Multiply 12 and \frac{1}{4} to get 3.

$$\left(-x\right)^{6}y^{12}z^{6}+8x^{6}y^{12}z^{6}-3x^{6}y^{8}z^{6}y^{4}$$

To multiply powers of the same base, add their exponents. Add 8 and 4 to get 12.

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

Combine 8x^{6}y^{12}z^{6} and -3x^{6}y^{12}z^{6} to get 5x^{6}y^{12}z^{6}.

$$\left(-x\right)^{6}y^{12}z^{6}+5x^{6}y^{12}z^{6}$$

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

$$\left(-1\right)^{6}x^{6}y^{12}z^{6}+5x^{6}y^{12}z^{6}$$

Calculate -1 to the power of 6 and get 1.

$$1x^{6}y^{12}z^{6}+5x^{6}y^{12}z^{6}$$

Combine 1x^{6}y^{12}z^{6} and 5x^{6}y^{12}z^{6} to get 6x^{6}y^{12}z^{6}.

$$6x^{6}y^{12}z^{6}$$

Expand

$$6y^{12}\left(xz\right)^{6}$$

Solution Steps

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

$$\left(\left(-x\right)y^{2}z\right)^{6}-\left(-2x^{2}y^{4}z^{2}\right)^{3}-3\left(-2x^{3}y^{4}z^{3}\right)^{2}\left(y^{2}-\frac{1}{2}y^{2}\right)^{2}$$

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

$$\left(-x\right)^{6}\left(y^{2}\right)^{6}z^{6}-\left(-2x^{2}y^{4}z^{2}\right)^{3}-3\left(-2x^{3}y^{4}z^{3}\right)^{2}\left(y^{2}-\frac{1}{2}y^{2}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 6 to get 12.

$$\left(-x\right)^{6}y^{12}z^{6}-\left(-2x^{2}y^{4}z^{2}\right)^{3}-3\left(-2x^{3}y^{4}z^{3}\right)^{2}\left(y^{2}-\frac{1}{2}y^{2}\right)^{2}$$

Expand \left(-2x^{2}y^{4}z^{2}\right)^{3}.

$$\left(-x\right)^{6}y^{12}z^{6}-\left(-2\right)^{3}\left(x^{2}\right)^{3}\left(y^{4}\right)^{3}\left(z^{2}\right)^{3}-3\left(-2x^{3}y^{4}z^{3}\right)^{2}\left(y^{2}-\frac{1}{2}y^{2}\right)^{2}$$

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

$$\left(-x\right)^{6}y^{12}z^{6}-\left(-2\right)^{3}x^{6}\left(y^{4}\right)^{3}\left(z^{2}\right)^{3}-3\left(-2x^{3}y^{4}z^{3}\right)^{2}\left(y^{2}-\frac{1}{2}y^{2}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 3 to get 12.

$$\left(-x\right)^{6}y^{12}z^{6}-\left(-2\right)^{3}x^{6}y^{12}\left(z^{2}\right)^{3}-3\left(-2x^{3}y^{4}z^{3}\right)^{2}\left(y^{2}-\frac{1}{2}y^{2}\right)^{2}$$

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

$$\left(-x\right)^{6}y^{12}z^{6}-\left(-2\right)^{3}x^{6}y^{12}z^{6}-3\left(-2x^{3}y^{4}z^{3}\right)^{2}\left(y^{2}-\frac{1}{2}y^{2}\right)^{2}$$

Calculate -2 to the power of 3 and get -8.

$$\left(-x\right)^{6}y^{12}z^{6}-\left(-8x^{6}y^{12}z^{6}\right)-3\left(-2x^{3}y^{4}z^{3}\right)^{2}\left(y^{2}-\frac{1}{2}y^{2}\right)^{2}$$

The opposite of -8x^{6}y^{12}z^{6} is 8x^{6}y^{12}z^{6}.

$$\left(-x\right)^{6}y^{12}z^{6}+8x^{6}y^{12}z^{6}-3\left(-2x^{3}y^{4}z^{3}\right)^{2}\left(y^{2}-\frac{1}{2}y^{2}\right)^{2}$$

Expand \left(-2x^{3}y^{4}z^{3}\right)^{2}.

$$\left(-x\right)^{6}y^{12}z^{6}+8x^{6}y^{12}z^{6}-3\left(-2\right)^{2}\left(x^{3}\right)^{2}\left(y^{4}\right)^{2}\left(z^{3}\right)^{2}\left(y^{2}-\frac{1}{2}y^{2}\right)^{2}$$

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

$$\left(-x\right)^{6}y^{12}z^{6}+8x^{6}y^{12}z^{6}-3\left(-2\right)^{2}x^{6}\left(y^{4}\right)^{2}\left(z^{3}\right)^{2}\left(y^{2}-\frac{1}{2}y^{2}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 2 to get 8.

$$\left(-x\right)^{6}y^{12}z^{6}+8x^{6}y^{12}z^{6}-3\left(-2\right)^{2}x^{6}y^{8}\left(z^{3}\right)^{2}\left(y^{2}-\frac{1}{2}y^{2}\right)^{2}$$

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

$$\left(-x\right)^{6}y^{12}z^{6}+8x^{6}y^{12}z^{6}-3\left(-2\right)^{2}x^{6}y^{8}z^{6}\left(y^{2}-\frac{1}{2}y^{2}\right)^{2}$$

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

$$\left(-x\right)^{6}y^{12}z^{6}+8x^{6}y^{12}z^{6}-3\times 4x^{6}y^{8}z^{6}\left(y^{2}-\frac{1}{2}y^{2}\right)^{2}$$

Multiply 3 and 4 to get 12.

$$\left(-x\right)^{6}y^{12}z^{6}+8x^{6}y^{12}z^{6}-12x^{6}y^{8}z^{6}\left(y^{2}-\frac{1}{2}y^{2}\right)^{2}$$

Combine y^{2} and -\frac{1}{2}y^{2} to get \frac{1}{2}y^{2}.

$$\left(-x\right)^{6}y^{12}z^{6}+8x^{6}y^{12}z^{6}-12x^{6}y^{8}z^{6}\times \left(\frac{1}{2}y^{2}\right)^{2}$$

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

$$\left(-x\right)^{6}y^{12}z^{6}+8x^{6}y^{12}z^{6}-12x^{6}y^{8}z^{6}\times \left(\frac{1}{2}\right)^{2}\left(y^{2}\right)^{2}$$

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

$$\left(-x\right)^{6}y^{12}z^{6}+8x^{6}y^{12}z^{6}-12x^{6}y^{8}z^{6}\times \left(\frac{1}{2}\right)^{2}y^{4}$$

Calculate \frac{1}{2} to the power of 2 and get \frac{1}{4}.

$$\left(-x\right)^{6}y^{12}z^{6}+8x^{6}y^{12}z^{6}-12x^{6}y^{8}z^{6}\times \frac{1}{4}y^{4}$$

Multiply 12 and \frac{1}{4} to get 3.

$$\left(-x\right)^{6}y^{12}z^{6}+8x^{6}y^{12}z^{6}-3x^{6}y^{8}z^{6}y^{4}$$

To multiply powers of the same base, add their exponents. Add 8 and 4 to get 12.

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

Combine 8x^{6}y^{12}z^{6} and -3x^{6}y^{12}z^{6} to get 5x^{6}y^{12}z^{6}.

$$\left(-x\right)^{6}y^{12}z^{6}+5x^{6}y^{12}z^{6}$$

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

$$\left(-1\right)^{6}x^{6}y^{12}z^{6}+5x^{6}y^{12}z^{6}$$

Calculate -1 to the power of 6 and get 1.

$$1x^{6}y^{12}z^{6}+5x^{6}y^{12}z^{6}$$

Combine 1x^{6}y^{12}z^{6} and 5x^{6}y^{12}z^{6} to get 6x^{6}y^{12}z^{6}.

$$6x^{6}y^{12}z^{6}$$