Solve n(5^3)^2*(5^2)^3

$$n ( 5 ^ { 3 } ) ^ { 2 } \times ( 5 ^ { 2 } ) ^ { 3 }$$

Evaluate

$$244140625n$$

Solution Steps

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

$$n\times 5^{6}\times \left(5^{2}\right)^{3}$$

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

$$n\times 5^{6}\times 5^{6}$$

To multiply powers of the same base, add their exponents. Add 6 and 6 to get 12.

$$n\times 5^{12}$$

Calculate 5 to the power of 12 and get 244140625.

$$n\times 244140625$$

Differentiate w.r.t. n

$$244140625$$

Steps Using Definition of a Derivative

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

$$\frac{\mathrm{d}}{\mathrm{d}n}(n\times 5^{6}\times \left(5^{2}\right)^{3})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}n}(n\times 5^{6}\times 5^{6})$$

To multiply powers of the same base, add their exponents. Add 6 and 6 to get 12.

$$\frac{\mathrm{d}}{\mathrm{d}n}(n\times 5^{12})$$

Calculate 5 to the power of 12 and get 244140625.

$$\frac{\mathrm{d}}{\mathrm{d}n}(n\times 244140625)$$

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

$$244140625n^{1-1}$$

Subtract 1 from 1.

$$244140625n^{0}$$

For any term t except 0, t^{0}=1.

$$244140625\times 1$$

For any term t, t\times 1=t and 1t=t.

$$244140625$$