Solve frac{(2ca^{-5}b^{4})^{-3}(ab)^{-1}}{(3a^3)^2(4b^-9c)^0}

$$\frac { ( 2 c a ^ { - 5 } b ^ { 4 } ) ^ { - 3 } ( a b ) ^ { - 1 } } { ( 3 a ^ { 3 } ) ^ { 2 } ( 4 b ^ { - 9 } c ) ^ { 0 } }$$

Evaluate

$$\frac{a^{8}}{72c^{3}b^{13}}$$

Solution Steps

Expand \left(2ca^{-5}b^{4}\right)^{-3}.

$$\frac{2^{-3}c^{-3}\left(a^{-5}\right)^{-3}\left(b^{4}\right)^{-3}\left(ab\right)^{-1}}{\left(3a^{3}\right)^{2}\times \left(4b^{-9}c\right)^{0}}$$

To raise a power to another power, multiply the exponents. Multiply -5 and -3 to get 15.

$$\frac{2^{-3}c^{-3}a^{15}\left(b^{4}\right)^{-3}\left(ab\right)^{-1}}{\left(3a^{3}\right)^{2}\times \left(4b^{-9}c\right)^{0}}$$

To raise a power to another power, multiply the exponents. Multiply 4 and -3 to get -12.

$$\frac{2^{-3}c^{-3}a^{15}b^{-12}\left(ab\right)^{-1}}{\left(3a^{3}\right)^{2}\times \left(4b^{-9}c\right)^{0}}$$

Calculate 2 to the power of -3 and get \frac{1}{8}.

$$\frac{\frac{1}{8}c^{-3}a^{15}b^{-12}\left(ab\right)^{-1}}{\left(3a^{3}\right)^{2}\times \left(4b^{-9}c\right)^{0}}$$

Expand \left(ab\right)^{-1}.

$$\frac{\frac{1}{8}c^{-3}a^{15}b^{-12}a^{-1}b^{-1}}{\left(3a^{3}\right)^{2}\times \left(4b^{-9}c\right)^{0}}$$

To multiply powers of the same base, add their exponents. Add 15 and -1 to get 14.

$$\frac{\frac{1}{8}c^{-3}a^{14}b^{-12}b^{-1}}{\left(3a^{3}\right)^{2}\times \left(4b^{-9}c\right)^{0}}$$

To multiply powers of the same base, add their exponents. Add -12 and -1 to get -13.

$$\frac{\frac{1}{8}c^{-3}a^{14}b^{-13}}{\left(3a^{3}\right)^{2}\times \left(4b^{-9}c\right)^{0}}$$

Expand \left(3a^{3}\right)^{2}.

$$\frac{\frac{1}{8}c^{-3}a^{14}b^{-13}}{3^{2}\left(a^{3}\right)^{2}\times \left(4b^{-9}c\right)^{0}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$\frac{\frac{1}{8}c^{-3}a^{14}b^{-13}}{3^{2}a^{6}\times \left(4b^{-9}c\right)^{0}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{\frac{1}{8}c^{-3}a^{14}b^{-13}}{9a^{6}\times \left(4b^{-9}c\right)^{0}}$$

Calculate 4b^{-9}c to the power of 0 and get 1.

$$\frac{\frac{1}{8}c^{-3}a^{14}b^{-13}}{9a^{6}\times 1}$$

Multiply 9 and 1 to get 9.

$$\frac{\frac{1}{8}c^{-3}a^{14}b^{-13}}{9a^{6}}$$

Cancel out a^{6} in both numerator and denominator.

$$\frac{\frac{1}{8}b^{-13}c^{-3}a^{8}}{9}$$

Divide \frac{1}{8}b^{-13}c^{-3}a^{8} by 9 to get \frac{1}{72}b^{-13}c^{-3}a^{8}.

$$\frac{1}{72}b^{-13}c^{-3}a^{8}$$

Differentiate w.r.t. c

$$-\frac{a^{8}}{24c^{4}b^{13}}$$