Solve 3(a^{2})^{4}-(-a)*(a^{4})^{4}+(a^{4})^2*(-a^3)(a^2)^3

$$3 ( a ^ { 2 } ) ^ { 4 } - ( - a ) \cdot ( a ^ { 4 } ) ^ { 4 } + ( a ^ { 4 } ) ^ { 2 } \cdot ( - a ^ { 3 } ) ( a ^ { 2 } ) ^ { 3 }$$

Evaluate

$$3a^{8}$$

Solution Steps

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

$$3a^{8}-\left(-a\right)\left(a^{4}\right)^{4}+\left(a^{4}\right)^{2}\left(-a^{3}\right)\left(a^{2}\right)^{3}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 4 to get 16.

$$3a^{8}-\left(-a\right)a^{16}+\left(a^{4}\right)^{2}\left(-a^{3}\right)\left(a^{2}\right)^{3}$$

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

$$3a^{8}-\left(-a\right)a^{16}+a^{8}\left(-a^{3}\right)\left(a^{2}\right)^{3}$$

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

$$3a^{8}-\left(-a\right)a^{16}+a^{8}\left(-a^{3}\right)a^{6}$$

To multiply powers of the same base, add their exponents. Add 8 and 6 to get 14.

$$3a^{8}-\left(-a\right)a^{16}+a^{14}\left(-a^{3}\right)$$

To multiply powers of the same base, add their exponents. Add 1 and 16 to get 17.

$$3a^{8}-\left(-a^{17}\right)+a^{14}\left(-1\right)a^{3}$$

Multiply -1 and -1 to get 1.

$$3a^{8}+a^{17}+a^{14}\left(-1\right)a^{3}$$

To multiply powers of the same base, add their exponents. Add 14 and 3 to get 17.

$$3a^{8}+a^{17}+a^{17}\left(-1\right)$$

Combine a^{17} and a^{17}\left(-1\right) to get 0.

$$3a^{8}$$

Differentiate w.r.t. a

$$24a^{7}$$

Steps Using Definition of a Derivative

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(3a^{8}-\left(-a\right)\left(a^{4}\right)^{4}+\left(a^{4}\right)^{2}\left(-a^{3}\right)\left(a^{2}\right)^{3})$$

To raise a power to another power, multiply the exponents. Multiply 4 and 4 to get 16.

$$\frac{\mathrm{d}}{\mathrm{d}a}(3a^{8}-\left(-a\right)a^{16}+\left(a^{4}\right)^{2}\left(-a^{3}\right)\left(a^{2}\right)^{3})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(3a^{8}-\left(-a\right)a^{16}+a^{8}\left(-a^{3}\right)\left(a^{2}\right)^{3})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(3a^{8}-\left(-a\right)a^{16}+a^{8}\left(-a^{3}\right)a^{6})$$

To multiply powers of the same base, add their exponents. Add 8 and 6 to get 14.

$$\frac{\mathrm{d}}{\mathrm{d}a}(3a^{8}-\left(-a\right)a^{16}+a^{14}\left(-a^{3}\right))$$

To multiply powers of the same base, add their exponents. Add 1 and 16 to get 17.

$$\frac{\mathrm{d}}{\mathrm{d}a}(3a^{8}-\left(-a^{17}\right)+a^{14}\left(-1\right)a^{3})$$

Multiply -1 and -1 to get 1.

$$\frac{\mathrm{d}}{\mathrm{d}a}(3a^{8}+a^{17}+a^{14}\left(-1\right)a^{3})$$

To multiply powers of the same base, add their exponents. Add 14 and 3 to get 17.

$$\frac{\mathrm{d}}{\mathrm{d}a}(3a^{8}+a^{17}+a^{17}\left(-1\right))$$

Combine a^{17} and a^{17}\left(-1\right) to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}a}(3a^{8})$$

The derivative of ax^{n} is nax^{n-1}.

$$8\times 3a^{8-1}$$

Multiply 8 times 3.

$$24a^{8-1}$$

Subtract 1 from 8.

$$24a^{7}$$