Solve q^{-1}r^{-1}s^{5}t^{3}*q^{5}r^{-1}s^{0}t^-1*q^-9rs^0t^90

$$q ^ { - 1 } r ^ { - 1 } s ^ { 5 } t ^ { 3 } \cdot q ^ { 5 } r ^ { - 1 } s ^ { 0 } t ^ { - 1 } \cdot q ^ { - 9 } r s ^ { 0 } t ^ { 90 }$$

Evaluate

$$\frac{\left(\frac{s}{q}\right)^{5}t^{92}}{r}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add -1 and 5 to get 4.

$$q^{4}r^{-1}s^{5}t^{3}r^{-1}s^{0}t^{-1}q^{-9}rs^{0}t^{90}$$

To multiply powers of the same base, add their exponents. Add 4 and -9 to get -5.

$$q^{-5}r^{-1}s^{5}t^{3}r^{-1}s^{0}t^{-1}rs^{0}t^{90}$$

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

$$q^{-5}r^{-2}s^{5}t^{3}s^{0}t^{-1}rs^{0}t^{90}$$

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

$$q^{-5}r^{-1}s^{5}t^{3}s^{0}t^{-1}s^{0}t^{90}$$

To multiply powers of the same base, add their exponents. Add 5 and 0 to get 5.

$$q^{-5}r^{-1}s^{5}t^{3}t^{-1}s^{0}t^{90}$$

To multiply powers of the same base, add their exponents. Add 5 and 0 to get 5.

$$q^{-5}r^{-1}s^{5}t^{3}t^{-1}t^{90}$$

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

$$q^{-5}r^{-1}s^{5}t^{2}t^{90}$$

To multiply powers of the same base, add their exponents. Add 2 and 90 to get 92.

$$q^{-5}r^{-1}s^{5}t^{92}$$

Differentiate w.r.t. r

$$-\frac{\left(\frac{s}{q}\right)^{5}t^{92}}{r^{2}}$$