Solve (frac{r^{-1}s^{2}t^{-3}}{r^-2s^0t^1})^-1

$$( \frac { r ^ { - 1 } s ^ { 2 } t ^ { - 3 } } { r ^ { - 2 } s ^ { 0 } t ^ { 1 } } ) ^ { - 1 }$$

Evaluate

$$\frac{t^{4}}{rs^{2}}$$

Solution Steps

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

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

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

$$\left(\frac{r^{1}s^{2}}{t^{4}}\right)^{-1}$$

Calculate r to the power of 1 and get r.

$$\left(\frac{rs^{2}}{t^{4}}\right)^{-1}$$

To raise \frac{rs^{2}}{t^{4}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(rs^{2}\right)^{-1}}{\left(t^{4}\right)^{-1}}$$

To raise a power to another power, multiply the exponents. Multiply 4 and -1 to get -4.

$$\frac{\left(rs^{2}\right)^{-1}}{t^{-4}}$$

Expand \left(rs^{2}\right)^{-1}.

$$\frac{r^{-1}\left(s^{2}\right)^{-1}}{t^{-4}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and -1 to get -2.

$$\frac{r^{-1}s^{-2}}{t^{-4}}$$

Differentiate w.r.t. r

$$-\frac{t^{4}}{\left(rs\right)^{2}}$$