Solve (n^0)^2*(2n^4)^3

$$( n ^ { 0 } ) ^ { 2 } \cdot ( 2 n ^ { 4 } ) ^ { 3 }$$

Evaluate

$$8n^{12}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(n^{0}\right)^{2}\times \left(2n^{4}\right)^{3}$$

To raise the product of two or more numbers to a power, raise each number to the power and take their product.

$$1^{2}\left(n^{0}\right)^{2}\times 2^{3}\left(n^{4}\right)^{3}$$

Use the Commutative Property of Multiplication.

$$1^{2}\times 2^{3}\left(n^{0}\right)^{2}\left(n^{4}\right)^{3}$$

To raise a power to another power, multiply the exponents.

$$1^{2}\times 2^{3}n^{0}n^{4\times 3}$$

Multiply 4 times 3.

$$1^{2}\times 2^{3}n^{0}n^{12}$$

To multiply powers of the same base, add their exponents.

$$1^{2}\times 2^{3}n^{12}$$

Raise 2 to the power 3.

$$8n^{12}$$

Differentiate w.r.t. n

$$96n^{11}$$

Steps Using Definition of a Derivative

To raise a power to another power, multiply the exponents. Multiply 0 and 2 to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}n}(n^{0}\times \left(2n^{4}\right)^{3})$$

Calculate n to the power of 0 and get 1.

$$\frac{\mathrm{d}}{\mathrm{d}n}(1\times \left(2n^{4}\right)^{3})$$

Expand \left(2n^{4}\right)^{3}.

$$\frac{\mathrm{d}}{\mathrm{d}n}(1\times 2^{3}\left(n^{4}\right)^{3})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}n}(1\times 2^{3}n^{12})$$

Calculate 2 to the power of 3 and get 8.

$$\frac{\mathrm{d}}{\mathrm{d}n}(1\times 8n^{12})$$

Multiply 1 and 8 to get 8.

$$\frac{\mathrm{d}}{\mathrm{d}n}(8n^{12})$$

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

$$12\times 8n^{12-1}$$

Multiply 12 times 8.

$$96n^{12-1}$$

Subtract 1 from 12.

$$96n^{11}$$