Solve 3ps/4pqrdiv3pq^2/12p^3q*14s^3/7qr

$$\frac { 3 p s } { 4 p q r } \div \frac { 3 p q ^ { 2 } } { 12 p ^ { 3 } q } \times \frac { 14 s ^ { 3 } } { 7 q r }$$

Evaluate

$$\frac{6\times \left(\frac{p}{r}\right)^{2}s^{4}}{q^{3}}$$

Solution Steps

Divide \frac{3ps}{4pqr} by \frac{3pq^{2}}{12p^{3}q} by multiplying \frac{3ps}{4pqr} by the reciprocal of \frac{3pq^{2}}{12p^{3}q}.

$$\frac{3ps\times 12p^{3}q}{4pqr\times 3pq^{2}}\times \frac{14s^{3}}{7qr}$$

Cancel out 3\times 4ppq in both numerator and denominator.

$$\frac{3sp^{2}}{rq^{2}}\times \frac{14s^{3}}{7qr}$$

Cancel out 7 in both numerator and denominator.

$$\frac{3sp^{2}}{rq^{2}}\times \frac{2s^{3}}{qr}$$

Multiply \frac{3sp^{2}}{rq^{2}} times \frac{2s^{3}}{qr} by multiplying numerator times numerator and denominator times denominator.

$$\frac{3sp^{2}\times 2s^{3}}{rq^{2}qr}$$

To multiply powers of the same base, add their exponents. Add 1 and 3 to get 4.

$$\frac{3s^{4}p^{2}\times 2}{rq^{2}qr}$$

Multiply r and r to get r^{2}.

$$\frac{3s^{4}p^{2}\times 2}{r^{2}q^{2}q}$$

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

$$\frac{3s^{4}p^{2}\times 2}{r^{2}q^{3}}$$

Multiply 3 and 2 to get 6.

$$\frac{6s^{4}p^{2}}{r^{2}q^{3}}$$

Differentiate w.r.t. r

$$-\frac{12p^{2}s^{4}}{\left(qr\right)^{3}}$$