Solve frac{(z^{3})^{-2}(z^5)^4}{(z^4)^2}

$$\frac { ( z ^ { 3 } ) ^ { - 2 } ( z ^ { 5 } ) ^ { 4 } } { ( z ^ { 4 } ) ^ { 2 } }$$

Evaluate

$$z^{6}$$

Solution Steps

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

$$\frac{z^{-6}\left(z^{5}\right)^{4}}{\left(z^{4}\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply 5 and 4 to get 20.

$$\frac{z^{-6}z^{20}}{\left(z^{4}\right)^{2}}$$

To multiply powers of the same base, add their exponents. Add -6 and 20 to get 14.

$$\frac{z^{14}}{\left(z^{4}\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 2 to get 8.

$$\frac{z^{14}}{z^{8}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract 8 from 14 to get 6.

$$z^{6}$$

Differentiate w.r.t. z

$$6z^{5}$$

Steps Using Definition of a Derivative

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

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{z^{-6}\left(z^{5}\right)^{4}}{\left(z^{4}\right)^{2}})$$

To raise a power to another power, multiply the exponents. Multiply 5 and 4 to get 20.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{z^{-6}z^{20}}{\left(z^{4}\right)^{2}})$$

To multiply powers of the same base, add their exponents. Add -6 and 20 to get 14.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{z^{14}}{\left(z^{4}\right)^{2}})$$

To raise a power to another power, multiply the exponents. Multiply 4 and 2 to get 8.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{z^{14}}{z^{8}})$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract 8 from 14 to get 6.

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

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

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

Subtract 1 from 6.

$$6z^{5}$$