Solve frac{r^{-1}s^{-1}t}{r^{-1}s^-1t^-1*r^7st^-2}

$$\frac { r ^ { - 1 } s ^ { - 1 } t } { r ^ { - 1 } s ^ { - 1 } t ^ { - 1 } \cdot r ^ { 7 } s t ^ { - 2 } }$$

Evaluate

$$\frac{t^{4}}{sr^{7}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add -1 and 7 to get 6.

$$\frac{r^{-1}s^{-1}t}{r^{6}s^{-1}t^{-1}st^{-2}}$$

Multiply s^{-1} and s to get 1.

$$\frac{r^{-1}s^{-1}t}{r^{6}t^{-1}t^{-2}}$$

To multiply powers of the same base, add their exponents. Add -1 and -2 to get -3.

$$\frac{r^{-1}s^{-1}t}{r^{6}t^{-3}}$$

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

$$\frac{\frac{1}{r}\times \frac{1}{s}t^{4}}{r^{6}}$$

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

$$\frac{\frac{1}{s}t^{4}}{r^{7}}$$

Express \frac{1}{s}t^{4} as a single fraction.

$$\frac{\frac{t^{4}}{s}}{r^{7}}$$

Express \frac{\frac{t^{4}}{s}}{r^{7}} as a single fraction.

$$\frac{t^{4}}{sr^{7}}$$

Differentiate w.r.t. s

$$-\frac{t^{4}}{s^{2}r^{7}}$$