Solve (-4a^2b^3):(-8/5a^3b)

$$( - 4 a ^ { 2 } b ^ { 3 } ) : ( - \frac { 8 } { 5 } a ^ { 3 } b )$$

Evaluate

$$\frac{5b^{2}}{2a}$$

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{\left(-4\right)^{1}a^{2}b^{3}}{\left(-\frac{8}{5}\right)^{1}a^{3}b^{1}}$$

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

$$\frac{\left(-4\right)^{1}}{\left(-\frac{8}{5}\right)^{1}}a^{2-3}b^{3-1}$$

Subtract 3 from 2.

$$\frac{\left(-4\right)^{1}}{\left(-\frac{8}{5}\right)^{1}}\times \frac{1}{a}b^{3-1}$$

Subtract 1 from 3.

$$\frac{\left(-4\right)^{1}}{\left(-\frac{8}{5}\right)^{1}}\times \frac{1}{a}b^{2}$$

Divide -4 by -\frac{8}{5} by multiplying -4 by the reciprocal of -\frac{8}{5}.

$$\frac{5}{2}\times \frac{1}{a}b^{2}$$

Differentiate w.r.t. a

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

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\left(-\frac{4b^{3}}{-\frac{8b}{5}}\right)a^{2-3})$$

Do the arithmetic.

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

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}.

$$-\frac{5b^{2}}{2}a^{-1-1}$$

Do the arithmetic.

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