Solve frac{p^{4}r^-2s}{r^5s^-7}:rp/s^-4

$$\frac { p ^ { 4 } r ^ { - 2 } s } { r ^ { 5 } s ^ { - 7 } } : \frac { r p } { s ^ { - 4 } }$$

Evaluate

$$\frac{p^{3}s^{4}}{r^{8}}$$

Solution Steps

Divide \frac{p^{4}r^{-2}s}{r^{5}s^{-7}} by \frac{rp}{s^{-4}} by multiplying \frac{p^{4}r^{-2}s}{r^{5}s^{-7}} by the reciprocal of \frac{rp}{s^{-4}}.

$$\frac{p^{4}r^{-2}ss^{-4}}{r^{5}s^{-7}rp}$$

Cancel out p in both numerator and denominator.

$$\frac{s^{-4}r^{-2}sp^{3}}{s^{-7}rr^{5}}$$

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

$$\frac{s^{-4}r^{-2}p^{3}s^{8}}{rr^{5}}$$

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

$$\frac{s^{-4}p^{3}s^{8}}{rr^{7}}$$

To multiply powers of the same base, add their exponents. Add -4 and 8 to get 4.

$$\frac{s^{4}p^{3}}{rr^{7}}$$

To multiply powers of the same base, add their exponents. Add 1 and 7 to get 8.

$$\frac{s^{4}p^{3}}{r^{8}}$$

Differentiate w.r.t. p

$$\frac{3p^{2}s^{4}}{r^{8}}$$