Solve -4m^5n^2/6m^2n^6

$$\frac { - 4 m ^ { 5 } n ^ { 2 } } { 6 m ^ { 2 } n ^ { 6 } }$$

Evaluate

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

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{\left(-4\right)^{1}m^{5}n^{2}}{6^{1}m^{2}n^{6}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\frac{\left(-4\right)^{1}}{6^{1}}m^{5-2}n^{2-6}$$

Subtract 2 from 5.

$$\frac{\left(-4\right)^{1}}{6^{1}}m^{3}n^{2-6}$$

Subtract 6 from 2.

$$\frac{\left(-4\right)^{1}}{6^{1}}m^{3}n^{-4}$$

Reduce the fraction \frac{-4}{6} to lowest terms by extracting and canceling out 2.

$$-\frac{2}{3}m^{3}\times \frac{1}{n^{4}}$$

Differentiate w.r.t. m

$$-\frac{2m^{2}}{n^{4}}$$

Solution Steps

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

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

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\left(-\frac{2}{3n^{4}}\right)m^{3})$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$3\left(-\frac{2}{3n^{4}}\right)m^{3-1}$$

Do the arithmetic.

$$\left(-\frac{2}{n^{4}}\right)m^{2}$$