Solve q^{-1}r^{0}s^{6}t^{2}*q^{-1}rst^{0}*q^-9r^0s^-3t^-7

$$q ^ { - 1 } r ^ { 0 } s ^ { 6 } t ^ { 2 } \cdot q ^ { - 1 } r s t ^ { 0 } \cdot q ^ { - 9 } r ^ { 0 } s ^ { - 3 } t ^ { - 7 }$$

Evaluate

$$\frac{rs^{4}}{t^{5}q^{11}}$$

Solution Steps

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

$$q^{-2}r^{0}s^{6}t^{2}rst^{0}q^{-9}r^{0}s^{-3}t^{-7}$$

To multiply powers of the same base, add their exponents. Add -2 and -9 to get -11.

$$q^{-11}r^{0}s^{6}t^{2}rst^{0}r^{0}s^{-3}t^{-7}$$

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

$$q^{-11}r^{1}s^{6}t^{2}st^{0}r^{0}s^{-3}t^{-7}$$

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

$$q^{-11}r^{1}s^{6}t^{2}st^{0}s^{-3}t^{-7}$$

To multiply powers of the same base, add their exponents. Add 6 and 1 to get 7.

$$q^{-11}r^{1}s^{7}t^{2}t^{0}s^{-3}t^{-7}$$

To multiply powers of the same base, add their exponents. Add 7 and -3 to get 4.

$$q^{-11}r^{1}s^{4}t^{2}t^{0}t^{-7}$$

To multiply powers of the same base, add their exponents. Add 2 and 0 to get 2.

$$q^{-11}r^{1}s^{4}t^{2}t^{-7}$$

To multiply powers of the same base, add their exponents. Add 2 and -7 to get -5.

$$q^{-11}r^{1}s^{4}t^{-5}$$

Calculate r to the power of 1 and get r.

$$q^{-11}rs^{4}t^{-5}$$

Differentiate w.r.t. t

$$-\frac{5rs^{4}}{t^{6}q^{11}}$$