Solve 32s^2t^2/-40s^3t

$$\frac { 32 s ^ { 2 } t ^ { 2 } } { - 40 s ^ { 3 } t }$$

Evaluate

$$-\frac{4t}{5s}$$

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{32^{1}s^{2}t^{2}}{\left(-40\right)^{1}s^{3}t^{1}}$$

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

$$\frac{32^{1}}{\left(-40\right)^{1}}s^{2-3}t^{2-1}$$

Subtract 3 from 2.

$$\frac{32^{1}}{\left(-40\right)^{1}}\times \frac{1}{s}t^{2-1}$$

Subtract 1 from 2.

$$\frac{32^{1}}{\left(-40\right)^{1}}\times \frac{1}{s}t^{1}$$

Reduce the fraction \frac{32}{-40} to lowest terms by extracting and canceling out 8.

$$-\frac{4}{5}\times \frac{1}{s}t$$

Differentiate w.r.t. s

$$\frac{4t}{5s^{2}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}s}(\frac{32t^{2}}{-40t}s^{2-3})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}s}(\left(-\frac{4t}{5}\right)\times \frac{1}{s})$$

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}.

$$-\left(-\frac{4t}{5}\right)s^{-1-1}$$

Do the arithmetic.

$$\frac{4t}{5}s^{-2}$$