Solve (frac{Q^{-1}R^{-1}S^-2}{R^-5SQ^-8})^-1

$$( \frac { Q ^ { - 1 } R ^ { - 1 } S ^ { - 2 } } { R ^ { - 5 } S Q ^ { - 8 } } ) ^ { - 1 }$$

Evaluate

$$\frac{S^{3}}{R^{4}Q^{7}}$$

Solution Steps

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

$$\left(\frac{S^{-2}R^{4}Q^{7}}{S}\right)^{-1}$$

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

$$\left(\frac{R^{4}Q^{7}}{S^{3}}\right)^{-1}$$

To raise \frac{R^{4}Q^{7}}{S^{3}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(R^{4}Q^{7}\right)^{-1}}{\left(S^{3}\right)^{-1}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and -1 to get -3.

$$\frac{\left(R^{4}Q^{7}\right)^{-1}}{S^{-3}}$$

Expand \left(R^{4}Q^{7}\right)^{-1}.

$$\frac{\left(R^{4}\right)^{-1}\left(Q^{7}\right)^{-1}}{S^{-3}}$$

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

$$\frac{R^{-4}\left(Q^{7}\right)^{-1}}{S^{-3}}$$

To raise a power to another power, multiply the exponents. Multiply 7 and -1 to get -7.

$$\frac{R^{-4}Q^{-7}}{S^{-3}}$$

Differentiate w.r.t. R

$$-\frac{4S^{3}}{R^{5}Q^{7}}$$