Solve z^2*z^-1*z^-3*z^5=

$$z ^ { 2 } \cdot z ^ { - 1 } \cdot z ^ { - 3 } \cdot z ^ { 5 } =$$

Evaluate

$$z^{3}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 2 and -1 to get 1.

$$z^{1}z^{-3}z^{5}$$

To multiply powers of the same base, add their exponents. Add 1 and -3 to get -2.

$$z^{-2}z^{5}$$

To multiply powers of the same base, add their exponents. Add -2 and 5 to get 3.

$$z^{3}$$

Differentiate w.r.t. z

$$3z^{2}$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add 2 and -1 to get 1.

$$\frac{\mathrm{d}}{\mathrm{d}z}(z^{1}z^{-3}z^{5})$$

To multiply powers of the same base, add their exponents. Add 1 and -3 to get -2.

$$\frac{\mathrm{d}}{\mathrm{d}z}(z^{-2}z^{5})$$

To multiply powers of the same base, add their exponents. Add -2 and 5 to get 3.

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

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

$$3z^{3-1}$$

Subtract 1 from 3.

$$3z^{2}$$