Solve 20z^2a^3/12z^3a

$$\frac { 20 z ^ { 2 } a ^ { 3 } } { 12 z ^ { 3 } a }$$

Evaluate

$$\frac{5a^{2}}{3z}$$

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{20^{1}z^{2}a^{3}}{12^{1}z^{3}a^{1}}$$

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

$$\frac{20^{1}}{12^{1}}z^{2-3}a^{3-1}$$

Subtract 3 from 2.

$$\frac{20^{1}}{12^{1}}\times \frac{1}{z}a^{3-1}$$

Subtract 1 from 3.

$$\frac{20^{1}}{12^{1}}\times \frac{1}{z}a^{2}$$

Reduce the fraction \frac{20}{12} to lowest terms by extracting and canceling out 4.

$$\frac{5}{3}\times \frac{1}{z}a^{2}$$

Differentiate w.r.t. z

$$-\frac{5\times \left(\frac{a}{z}\right)^{2}}{3}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{20a^{3}}{12a}z^{2-3})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{5a^{2}}{3}\times \frac{1}{z})$$

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}.

$$-\frac{5a^{2}}{3}z^{-1-1}$$

Do the arithmetic.

$$\left(-\frac{5a^{2}}{3}\right)z^{-2}$$