Solve frac{(2r^{5}s^{-3})^{-2}(7r^{-3}s^{5})^0}{(-r^4s^-3)^3}

$$\frac { ( 2 r ^ { 5 } s ^ { - 3 } ) ^ { - 2 } ( 7 r ^ { - 3 } s ^ { 5 } ) ^ { 0 } } { ( - r ^ { 4 } s ^ { - 3 } ) ^ { 3 } }$$

Evaluate

$$-\frac{s^{15}}{4r^{22}}$$

Solution Steps

Expand \left(2r^{5}s^{-3}\right)^{-2}.

$$\frac{2^{-2}\left(r^{5}\right)^{-2}\left(s^{-3}\right)^{-2}\times \left(7r^{-3}s^{5}\right)^{0}}{\left(\left(-r^{4}\right)s^{-3}\right)^{3}}$$

To raise a power to another power, multiply the exponents. Multiply 5 and -2 to get -10.

$$\frac{2^{-2}r^{-10}\left(s^{-3}\right)^{-2}\times \left(7r^{-3}s^{5}\right)^{0}}{\left(\left(-r^{4}\right)s^{-3}\right)^{3}}$$

To raise a power to another power, multiply the exponents. Multiply -3 and -2 to get 6.

$$\frac{2^{-2}r^{-10}s^{6}\times \left(7r^{-3}s^{5}\right)^{0}}{\left(\left(-r^{4}\right)s^{-3}\right)^{3}}$$

Calculate 2 to the power of -2 and get \frac{1}{4}.

$$\frac{\frac{1}{4}r^{-10}s^{6}\times \left(7r^{-3}s^{5}\right)^{0}}{\left(\left(-r^{4}\right)s^{-3}\right)^{3}}$$

Calculate 7r^{-3}s^{5} to the power of 0 and get 1.

$$\frac{\frac{1}{4}r^{-10}s^{6}\times 1}{\left(\left(-r^{4}\right)s^{-3}\right)^{3}}$$

Multiply \frac{1}{4} and 1 to get \frac{1}{4}.

$$\frac{\frac{1}{4}r^{-10}s^{6}}{\left(\left(-r^{4}\right)s^{-3}\right)^{3}}$$

Expand \left(\left(-r^{4}\right)s^{-3}\right)^{3}.

$$\frac{\frac{1}{4}r^{-10}s^{6}}{\left(-r^{4}\right)^{3}\left(s^{-3}\right)^{3}}$$

To raise a power to another power, multiply the exponents. Multiply -3 and 3 to get -9.

$$\frac{\frac{1}{4}r^{-10}s^{6}}{\left(-r^{4}\right)^{3}s^{-9}}$$

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

$$\frac{\frac{1}{4}r^{-10}s^{15}}{\left(-r^{4}\right)^{3}}$$

Expand \left(-r^{4}\right)^{3}.

$$\frac{\frac{1}{4}r^{-10}s^{15}}{\left(-1\right)^{3}\left(r^{4}\right)^{3}}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 3 to get 12.

$$\frac{\frac{1}{4}r^{-10}s^{15}}{\left(-1\right)^{3}r^{12}}$$

Calculate -1 to the power of 3 and get -1.

$$\frac{\frac{1}{4}r^{-10}s^{15}}{-r^{12}}$$

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

$$\frac{\frac{1}{4}s^{15}}{-r^{22}}$$

Differentiate w.r.t. s

$$-\frac{15s^{14}}{4r^{22}}$$