Solve frac{q^{-67}r^{-46}s^0}{q^-69rs*q^-1r^0s}

$$\frac { q ^ { - 67 } r ^ { - 46 } s ^ { 0 } } { q ^ { - 69 } r s \cdot q ^ { - 1 } r ^ { 0 } s }$$

Evaluate

$$\frac{q^{3}}{s^{2}r^{47}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add -69 and -1 to get -70.

$$\frac{q^{-67}r^{-46}s^{0}}{q^{-70}rsr^{0}s}$$

To multiply powers of the same base, add their exponents. Add 1 and 0 to get 1.

$$\frac{q^{-67}r^{-46}s^{0}}{q^{-70}r^{1}ss}$$

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

$$\frac{q^{-67}r^{-46}s^{0}}{q^{-70}r^{1}s^{2}}$$

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

$$\frac{r^{-46}s^{0}q^{3}}{r^{1}s^{2}}$$

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

$$\frac{q^{3}}{s^{2}r^{47}}$$

Differentiate w.r.t. s

$$-\frac{2\times \left(\frac{q}{s}\right)^{3}}{r^{47}}$$