Solve (frac{9^{-1}r^{-1}s^-2}{r^-5sq^-8})^-1

$$( \frac { 9 ^ { - 1 } r ^ { - 1 } s ^ { - 2 } } { r ^ { - 5 } s q ^ { - 8 } } ) ^ { - 1 }$$

Evaluate

$$\frac{9s^{3}}{r^{4}q^{8}}$$

Solution Steps

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

$$\left(\frac{\frac{1}{9}s^{-2}r^{4}}{q^{-8}s}\right)^{-1}$$

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

$$\left(\frac{\frac{1}{9}r^{4}}{q^{-8}s^{3}}\right)^{-1}$$

To raise \frac{\frac{1}{9}r^{4}}{q^{-8}s^{3}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(\frac{1}{9}r^{4}\right)^{-1}}{\left(q^{-8}s^{3}\right)^{-1}}$$

Expand \left(\frac{1}{9}r^{4}\right)^{-1}.

$$\frac{\left(\frac{1}{9}\right)^{-1}\left(r^{4}\right)^{-1}}{\left(q^{-8}s^{3}\right)^{-1}}$$

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

$$\frac{\left(\frac{1}{9}\right)^{-1}r^{-4}}{\left(q^{-8}s^{3}\right)^{-1}}$$

Calculate \frac{1}{9} to the power of -1 and get 9.

$$\frac{9r^{-4}}{\left(q^{-8}s^{3}\right)^{-1}}$$

Expand \left(q^{-8}s^{3}\right)^{-1}.

$$\frac{9r^{-4}}{\left(q^{-8}\right)^{-1}\left(s^{3}\right)^{-1}}$$

To raise a power to another power, multiply the exponents. Multiply -8 and -1 to get 8.

$$\frac{9r^{-4}}{q^{8}\left(s^{3}\right)^{-1}}$$

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

$$\frac{9r^{-4}}{q^{8}s^{-3}}$$

Differentiate w.r.t. q

$$-\frac{72s^{3}}{r^{4}q^{9}}$$