Solve (1000r^9t^12/s^-3)^frac{1{3}}

$$( \frac { 1000 r ^ { 9 } t ^ { 12 } } { s ^ { - 3 } } ) ^ { \frac { 1 } { 3 } }$$

Evaluate

$$10sr^{3}t^{4}$$

Solution Steps

To raise \frac{1000r^{9}t^{12}}{s^{-3}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(1000r^{9}t^{12}\right)^{\frac{1}{3}}}{\left(s^{-3}\right)^{\frac{1}{3}}}$$

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

$$\frac{\left(1000r^{9}t^{12}\right)^{\frac{1}{3}}}{s^{-1}}$$

Expand \left(1000r^{9}t^{12}\right)^{\frac{1}{3}}.

$$\frac{1000^{\frac{1}{3}}\left(r^{9}\right)^{\frac{1}{3}}\left(t^{12}\right)^{\frac{1}{3}}}{s^{-1}}$$

To raise a power to another power, multiply the exponents. Multiply 9 and \frac{1}{3} to get 3.

$$\frac{1000^{\frac{1}{3}}r^{3}\left(t^{12}\right)^{\frac{1}{3}}}{s^{-1}}$$

To raise a power to another power, multiply the exponents. Multiply 12 and \frac{1}{3} to get 4.

$$\frac{1000^{\frac{1}{3}}r^{3}t^{4}}{s^{-1}}$$

Calculate 1000 to the power of \frac{1}{3} and get 10.

$$\frac{10r^{3}t^{4}}{s^{-1}}$$

Differentiate w.r.t. r

$$30sr^{2}t^{4}$$