Solve left(z^3right)^2*left(z^3right)^4

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

Evaluate

$$z^{18}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(z^{3}\right)^{2}\left(z^{3}\right)^{4}$$

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

$$z^{3\times 2}z^{3\times 4}$$

Multiply 3 times 2.

$$z^{6}z^{3\times 4}$$

Multiply 3 times 4.

$$z^{6}z^{12}$$

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

$$z^{6+12}$$

Add the exponents 6 and 12.

$$z^{18}$$

Differentiate w.r.t. z

$$18z^{17}$$

Steps Using Definition of a Derivative

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

$$\frac{\mathrm{d}}{\mathrm{d}z}(z^{6}\left(z^{3}\right)^{4})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}z}(z^{6}z^{12})$$

To multiply powers of the same base, add their exponents. Add 6 and 12 to get 18.

$$\frac{\mathrm{d}}{\mathrm{d}z}(z^{18})$$

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

$$18z^{18-1}$$

Subtract 1 from 18.

$$18z^{17}$$