Solve (frac{p^{3}q^{6}}{q^2})^frac{1{3}}*p^frac{1{2}}q^frac{3{2}}

$$( \frac { p ^ { 3 } q ^ { 6 } } { q ^ { 2 } } ) ^ { \frac { 1 } { 3 } } \times p ^ { \frac { 1 } { 2 } } q ^ { \frac { 3 } { 2 } }$$

Evaluate

$$p^{\frac{3}{2}}q^{\frac{17}{6}}$$

Solution Steps

Cancel out q^{2} in both numerator and denominator.

$$\left(p^{3}q^{4}\right)^{\frac{1}{3}}p^{\frac{1}{2}}q^{\frac{3}{2}}$$

Expand \left(p^{3}q^{4}\right)^{\frac{1}{3}}.

$$\left(p^{3}\right)^{\frac{1}{3}}\left(q^{4}\right)^{\frac{1}{3}}p^{\frac{1}{2}}q^{\frac{3}{2}}$$

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

$$p^{1}\left(q^{4}\right)^{\frac{1}{3}}p^{\frac{1}{2}}q^{\frac{3}{2}}$$

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

$$p^{1}q^{\frac{4}{3}}p^{\frac{1}{2}}q^{\frac{3}{2}}$$

Calculate p to the power of 1 and get p.

$$pq^{\frac{4}{3}}p^{\frac{1}{2}}q^{\frac{3}{2}}$$

To multiply powers of the same base, add their exponents. Add 1 and \frac{1}{2} to get \frac{3}{2}.

$$p^{\frac{3}{2}}q^{\frac{4}{3}}q^{\frac{3}{2}}$$

To multiply powers of the same base, add their exponents. Add \frac{4}{3} and \frac{3}{2} to get \frac{17}{6}.

$$p^{\frac{3}{2}}q^{\frac{17}{6}}$$

Differentiate w.r.t. p

$$\frac{3\sqrt{p}q^{\frac{17}{6}}}{2}$$