Solve 2m^2*4m^frac{3{2}}*4m^-2

$$2 m ^ { 2 } \times 4 m ^ { \frac { 3 } { 2 } } \times 4 m ^ { - 2 }$$

Evaluate

$$32m^{\frac{3}{2}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 2 and \frac{3}{2} to get \frac{7}{2}.

$$2m^{\frac{7}{2}}\times 4\times 4m^{-2}$$

To multiply powers of the same base, add their exponents. Add \frac{7}{2} and -2 to get \frac{3}{2}.

$$2m^{\frac{3}{2}}\times 4\times 4$$

Multiply 2 and 4 to get 8.

$$8m^{\frac{3}{2}}\times 4$$

Multiply 8 and 4 to get 32.

$$32m^{\frac{3}{2}}$$

Differentiate w.r.t. m

$$48\sqrt{m}$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add 2 and \frac{3}{2} to get \frac{7}{2}.

$$\frac{\mathrm{d}}{\mathrm{d}m}(2m^{\frac{7}{2}}\times 4\times 4m^{-2})$$

To multiply powers of the same base, add their exponents. Add \frac{7}{2} and -2 to get \frac{3}{2}.

$$\frac{\mathrm{d}}{\mathrm{d}m}(2m^{\frac{3}{2}}\times 4\times 4)$$

Multiply 2 and 4 to get 8.

$$\frac{\mathrm{d}}{\mathrm{d}m}(8m^{\frac{3}{2}}\times 4)$$

Multiply 8 and 4 to get 32.

$$\frac{\mathrm{d}}{\mathrm{d}m}(32m^{\frac{3}{2}})$$

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

$$\frac{3}{2}\times 32m^{\frac{3}{2}-1}$$

Multiply \frac{3}{2} times 32.

$$48m^{\frac{3}{2}-1}$$

Subtract 1 from \frac{3}{2}.

$$48\sqrt{m}$$