Solve (m^5)^6m/m^15

$$\frac { ( m ^ { 5 } ) ^ { 6 } m } { m ^ { 15 } }$$

Evaluate

$$m^{16}$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply 5 and 6 to get 30.

$$\frac{m^{30}m}{m^{15}}$$

To multiply powers of the same base, add their exponents. Add 30 and 1 to get 31.

$$\frac{m^{31}}{m^{15}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract 15 from 31 to get 16.

$$m^{16}$$

Differentiate w.r.t. m

$$16m^{15}$$

Steps Using Definition of a Derivative

To raise a power to another power, multiply the exponents. Multiply 5 and 6 to get 30.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{m^{30}m}{m^{15}})$$

To multiply powers of the same base, add their exponents. Add 30 and 1 to get 31.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{m^{31}}{m^{15}})$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract 15 from 31 to get 16.

$$\frac{\mathrm{d}}{\mathrm{d}m}(m^{16})$$

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

$$16m^{16-1}$$

Subtract 1 from 16.

$$16m^{15}$$