Solve (2m^3)^0(3m^6)^-1

$$( 2 m ^ { 3 } ) ^ { 0 } ( 3 m ^ { 6 } ) ^ { - 1 }$$

Evaluate

$$\frac{1}{3m^{6}}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(2m^{3}\right)^{0}\times \frac{1}{3m^{6}}$$

To raise the product of two or more numbers to a power, raise each number to the power and take their product.

$$2^{0}\left(m^{3}\right)^{0}\times \frac{1}{3}\times \frac{1}{m^{6}}$$

Use the Commutative Property of Multiplication.

$$2^{0}\times \frac{1}{3}\left(m^{3}\right)^{0}\times \frac{1}{m^{6}}$$

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

$$2^{0}\times \frac{1}{3}m^{0}m^{6\left(-1\right)}$$

Multiply 6 times -1.

$$2^{0}\times \frac{1}{3}m^{0}m^{-6}$$

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

$$2^{0}\times \frac{1}{3}m^{-6}$$

Raise 2 to the power 0.

$$\frac{1}{3}m^{-6}$$

Differentiate w.r.t. m

$$-\frac{2}{m^{7}}$$

Steps Using Definition of a Derivative

Calculate 2m^{3} to the power of 0 and get 1.

$$\frac{\mathrm{d}}{\mathrm{d}m}(1\times \left(3m^{6}\right)^{-1})$$

Expand \left(3m^{6}\right)^{-1}.

$$\frac{\mathrm{d}}{\mathrm{d}m}(1\times 3^{-1}\left(m^{6}\right)^{-1})$$

To raise a power to another power, multiply the exponents. Multiply 6 and -1 to get -6.

$$\frac{\mathrm{d}}{\mathrm{d}m}(1\times 3^{-1}m^{-6})$$

Calculate 3 to the power of -1 and get \frac{1}{3}.

$$\frac{\mathrm{d}}{\mathrm{d}m}(1\times \frac{1}{3}m^{-6})$$

Multiply 1 and \frac{1}{3} to get \frac{1}{3}.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{1}{3}m^{-6})$$

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

$$-6\times \frac{1}{3}m^{-6-1}$$

Multiply -6 times \frac{1}{3}.

$$-2m^{-6-1}$$

Subtract 1 from -6.

$$-2m^{-7}$$