Solve frac{-(5p^{2}q^-3)^2}{10pr^-4}*frac{2sqrt[3]{pr}}{q^3}

$$\frac { - ( 5 p ^ { 2 } q ^ { - 3 } ) ^ { 2 } } { 10 p r ^ { - 4 } } \times \frac { 2 \sqrt[ 3 ] { p r } } { q ^ { 3 } }$$

Evaluate

$$-\frac{5p^{\frac{10}{3}}r^{\frac{13}{3}}}{q^{9}}$$

Short Solution Steps

Expand \left(5p^{2}q^{-3}\right)^{2}.

$$\frac{-5^{2}\left(p^{2}\right)^{2}\left(q^{-3}\right)^{2}}{10pr^{-4}}\times \frac{2\sqrt[3]{pr}}{q^{3}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\frac{-5^{2}p^{4}\left(q^{-3}\right)^{2}}{10pr^{-4}}\times \frac{2\sqrt[3]{pr}}{q^{3}}$$

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

$$\frac{-5^{2}p^{4}q^{-6}}{10pr^{-4}}\times \frac{2\sqrt[3]{pr}}{q^{3}}$$

Calculate 5 to the power of 2 and get 25.

$$\frac{-25p^{4}q^{-6}}{10pr^{-4}}\times \frac{2\sqrt[3]{pr}}{q^{3}}$$

Multiply \frac{-25p^{4}q^{-6}}{10pr^{-4}} times \frac{2\sqrt[3]{pr}}{q^{3}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(-25p^{4}q^{-6}\right)\times 2\sqrt[3]{pr}}{10pr^{-4}q^{3}}$$

Cancel out 2 in both numerator and denominator.

$$\frac{\left(-25q^{-6}p^{4}\right)\sqrt[3]{pr}}{5r^{-4}pq^{3}}$$

Factor the expressions that are not already factored.

$$\frac{-25q^{-6}\sqrt[3]{pr}p^{4}}{5r^{-4}pq^{3}}$$

Cancel out 5p in both numerator and denominator.

$$\frac{-5q^{-6}\sqrt[3]{pr}p^{3}}{r^{-4}q^{3}}$$

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

$$\frac{-5\sqrt[3]{pr}p^{3}}{r^{-4}q^{9}}$$

Expand the expression.

$$\frac{-5\sqrt[3]{r}p^{\frac{10}{3}}}{r^{-4}q^{9}}$$

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

$$\frac{-5p^{\frac{10}{3}}r^{\frac{13}{3}}}{q^{9}}$$

Differentiate w.r.t. p

$$-\frac{50p^{\frac{7}{3}}r^{\frac{13}{3}}}{3q^{9}}$$