Solve frac{r^{5}s^{7}*16*t^{-6}}{2^4r^-6s^9t^7}

$$\frac { r ^ { 5 } s ^ { 7 } \cdot 16 \cdot t ^ { - 6 } } { 2 ^ { 4 } r ^ { - 6 } s ^ { 9 } t ^ { 7 } }$$

Evaluate

$$\frac{r^{11}}{s^{2}t^{13}}$$

Solution Steps

Cancel out s^{7} in both numerator and denominator.

$$\frac{16t^{-6}r^{5}}{2^{4}r^{-6}s^{2}t^{7}}$$

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

$$\frac{16t^{-6}r^{11}}{2^{4}s^{2}t^{7}}$$

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

$$\frac{16r^{11}}{2^{4}s^{2}t^{13}}$$

Calculate 2 to the power of 4 and get 16.

$$\frac{16r^{11}}{16s^{2}t^{13}}$$

Cancel out 16 in both numerator and denominator.

$$\frac{r^{11}}{s^{2}t^{13}}$$

Differentiate w.r.t. r

$$\frac{11r^{10}}{s^{2}t^{13}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}r}(\frac{16s^{7}}{t^{6}\times 16t^{7}s^{9}}r^{5-\left(-6\right)})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}r}(\frac{1}{s^{2}t^{13}}r^{11})$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$11\times \frac{1}{s^{2}t^{13}}r^{11-1}$$

Do the arithmetic.

$$\frac{11}{s^{2}t^{13}}r^{10}$$