Solve [-(-a^{3}x)^{2}]^{2}*{[(1/5a^{6}x^{9})^{5}]^{0}}^{3}-x*(1/2a^{3}x)^3*(-2a^3)+(-2a^3x)^4

$$[ - ( - a ^ { 3 } x ) ^ { 2 } ] ^ { 2 } \cdot \{ [ ( \frac { 1 } { 5 } a ^ { 6 } x ^ { 9 } ) ^ { 5 } ] ^ { 0 } \} ^ { 3 } - x \cdot ( \frac { 1 } { 2 } a ^ { 3 } x ) ^ { 3 } \cdot ( - 2 a ^ { 3 } ) + ( - 2 a ^ { 3 } x ) ^ { 4 }$$

Evaluate

$$\frac{69x^{4}a^{12}}{4}$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply 5 and 0 to get 0.

$$\left(-\left(\left(-a^{3}\right)x\right)^{2}\right)^{2}\times \left(\left(\frac{1}{5}a^{6}x^{9}\right)^{0}\right)^{3}-x\times \left(\frac{1}{2}a^{3}x\right)^{3}\left(-2\right)a^{3}+\left(-2a^{3}x\right)^{4}$$

To raise a power to another power, multiply the exponents. Multiply 0 and 3 to get 0.

$$\left(-\left(\left(-a^{3}\right)x\right)^{2}\right)^{2}\times \left(\frac{1}{5}a^{6}x^{9}\right)^{0}-x\times \left(\frac{1}{2}a^{3}x\right)^{3}\left(-2\right)a^{3}+\left(-2a^{3}x\right)^{4}$$

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

$$\left(-\left(-a^{3}\right)^{2}x^{2}\right)^{2}\times \left(\frac{1}{5}a^{6}x^{9}\right)^{0}-x\times \left(\frac{1}{2}a^{3}x\right)^{3}\left(-2\right)a^{3}+\left(-2a^{3}x\right)^{4}$$

Calculate -a^{3} to the power of 2 and get \left(a^{3}\right)^{2}.

$$\left(-\left(a^{3}\right)^{2}x^{2}\right)^{2}\times \left(\frac{1}{5}a^{6}x^{9}\right)^{0}-x\times \left(\frac{1}{2}a^{3}x\right)^{3}\left(-2\right)a^{3}+\left(-2a^{3}x\right)^{4}$$

Calculate -\left(a^{3}\right)^{2}x^{2} to the power of 2 and get \left(\left(a^{3}\right)^{2}x^{2}\right)^{2}.

$$\left(\left(a^{3}\right)^{2}x^{2}\right)^{2}\times \left(\frac{1}{5}a^{6}x^{9}\right)^{0}-x\times \left(\frac{1}{2}a^{3}x\right)^{3}\left(-2\right)a^{3}+\left(-2a^{3}x\right)^{4}$$

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

$$\left(\left(a^{3}\right)^{2}\right)^{2}\left(x^{2}\right)^{2}\times \left(\frac{1}{5}a^{6}x^{9}\right)^{0}-x\times \left(\frac{1}{2}a^{3}x\right)^{3}\left(-2\right)a^{3}+\left(-2a^{3}x\right)^{4}$$

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

$$\left(a^{6}\right)^{2}\left(x^{2}\right)^{2}\times \left(\frac{1}{5}a^{6}x^{9}\right)^{0}-x\times \left(\frac{1}{2}a^{3}x\right)^{3}\left(-2\right)a^{3}+\left(-2a^{3}x\right)^{4}$$

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

$$a^{12}\left(x^{2}\right)^{2}\times \left(\frac{1}{5}a^{6}x^{9}\right)^{0}-x\times \left(\frac{1}{2}a^{3}x\right)^{3}\left(-2\right)a^{3}+\left(-2a^{3}x\right)^{4}$$

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

$$a^{12}x^{4}\times \left(\frac{1}{5}a^{6}x^{9}\right)^{0}-x\times \left(\frac{1}{2}a^{3}x\right)^{3}\left(-2\right)a^{3}+\left(-2a^{3}x\right)^{4}$$

Calculate \frac{1}{5}a^{6}x^{9} to the power of 0 and get 1.

$$a^{12}x^{4}\times 1-x\times \left(\frac{1}{2}a^{3}x\right)^{3}\left(-2\right)a^{3}+\left(-2a^{3}x\right)^{4}$$

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

$$a^{12}x^{4}\times 1-x\times \left(\frac{1}{2}\right)^{3}\left(a^{3}\right)^{3}x^{3}\left(-2\right)a^{3}+\left(-2a^{3}x\right)^{4}$$

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

$$a^{12}x^{4}\times 1-x\times \left(\frac{1}{2}\right)^{3}a^{9}x^{3}\left(-2\right)a^{3}+\left(-2a^{3}x\right)^{4}$$

Calculate \frac{1}{2} to the power of 3 and get \frac{1}{8}.

$$a^{12}x^{4}\times 1-x\times \frac{1}{8}a^{9}x^{3}\left(-2\right)a^{3}+\left(-2a^{3}x\right)^{4}$$

To multiply powers of the same base, add their exponents. Add 1 and 3 to get 4.

$$a^{12}x^{4}\times 1-x^{4}\times \frac{1}{8}a^{9}\left(-2\right)a^{3}+\left(-2a^{3}x\right)^{4}$$

Multiply \frac{1}{8} and -2 to get -\frac{1}{4}.

$$a^{12}x^{4}\times 1-x^{4}\left(-\frac{1}{4}\right)a^{9}a^{3}+\left(-2a^{3}x\right)^{4}$$

To multiply powers of the same base, add their exponents. Add 9 and 3 to get 12.

$$a^{12}x^{4}\times 1-x^{4}\left(-\frac{1}{4}\right)a^{12}+\left(-2a^{3}x\right)^{4}$$

Combine a^{12}x^{4}\times 1 and -x^{4}\left(-\frac{1}{4}\right)a^{12} to get \frac{5}{4}a^{12}x^{4}.

$$\frac{5}{4}a^{12}x^{4}+\left(-2a^{3}x\right)^{4}$$

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

$$\frac{5}{4}a^{12}x^{4}+\left(-2\right)^{4}\left(a^{3}\right)^{4}x^{4}$$

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

$$\frac{5}{4}a^{12}x^{4}+\left(-2\right)^{4}a^{12}x^{4}$$

Calculate -2 to the power of 4 and get 16.

$$\frac{5}{4}a^{12}x^{4}+16a^{12}x^{4}$$

Combine \frac{5}{4}a^{12}x^{4} and 16a^{12}x^{4} to get \frac{69}{4}a^{12}x^{4}.

$$\frac{69}{4}a^{12}x^{4}$$

Differentiate w.r.t. x

$$69x^{3}a^{12}$$