Solve (z^3)^2*(3^2)^3

$$( z ^ { 3 } ) ^ { 2 } \times ( 3 ^ { 2 } ) ^ { 3 }$$

Evaluate

$$729z^{6}$$

Solution Steps

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

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

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

$$z^{6}\times 3^{6}$$

Calculate 3 to the power of 6 and get 729.

$$z^{6}\times 729$$

Differentiate w.r.t. z

$$4374z^{5}$$

Steps Using Definition of a Derivative

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

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

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

$$\frac{\mathrm{d}}{\mathrm{d}z}(z^{6}\times 3^{6})$$

Calculate 3 to the power of 6 and get 729.

$$\frac{\mathrm{d}}{\mathrm{d}z}(z^{6}\times 729)$$

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

$$6\times 729z^{6-1}$$

Multiply 6 times 729.

$$4374z^{6-1}$$

Subtract 1 from 6.

$$4374z^{5}$$