Solve (frac{2a^{-4}b^{3}c^{3}}{6a^3b^-2c^-5})^-1

$$( \frac { 2 a ^ { - 4 } b ^ { 3 } c ^ { 3 } } { 6 a ^ { 3 } b ^ { - 2 } c ^ { - 5 } } ) ^ { - 1 }$$

Evaluate

$$\frac{3a^{7}}{b^{5}c^{8}}$$

Solution Steps

Cancel out 2 in both numerator and denominator.

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

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

$$\left(\frac{a^{-4}b^{5}c^{8}}{3a^{3}}\right)^{-1}$$

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

$$\left(\frac{b^{5}c^{8}}{3a^{7}}\right)^{-1}$$

To raise \frac{b^{5}c^{8}}{3a^{7}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(b^{5}c^{8}\right)^{-1}}{\left(3a^{7}\right)^{-1}}$$

Expand \left(b^{5}c^{8}\right)^{-1}.

$$\frac{\left(b^{5}\right)^{-1}\left(c^{8}\right)^{-1}}{\left(3a^{7}\right)^{-1}}$$

To raise a power to another power, multiply the exponents. Multiply 5 and -1 to get -5.

$$\frac{b^{-5}\left(c^{8}\right)^{-1}}{\left(3a^{7}\right)^{-1}}$$

To raise a power to another power, multiply the exponents. Multiply 8 and -1 to get -8.

$$\frac{b^{-5}c^{-8}}{\left(3a^{7}\right)^{-1}}$$

Expand \left(3a^{7}\right)^{-1}.

$$\frac{b^{-5}c^{-8}}{3^{-1}\left(a^{7}\right)^{-1}}$$

To raise a power to another power, multiply the exponents. Multiply 7 and -1 to get -7.

$$\frac{b^{-5}c^{-8}}{3^{-1}a^{-7}}$$

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

$$\frac{b^{-5}c^{-8}}{\frac{1}{3}a^{-7}}$$

Differentiate w.r.t. c

$$-\frac{24a^{7}}{b^{5}c^{9}}$$