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

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

Evaluate

$$\frac{3\times \left(\frac{a}{c}\right)^{2}}{5b^{8}}$$

Solution Steps

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

$$\left(\frac{5a^{-2}b^{8}}{3c^{-2}}\right)^{-1}$$

To raise \frac{5a^{-2}b^{8}}{3c^{-2}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(5a^{-2}b^{8}\right)^{-1}}{\left(3c^{-2}\right)^{-1}}$$

Expand \left(5a^{-2}b^{8}\right)^{-1}.

$$\frac{5^{-1}\left(a^{-2}\right)^{-1}\left(b^{8}\right)^{-1}}{\left(3c^{-2}\right)^{-1}}$$

To raise a power to another power, multiply the exponents. Multiply -2 and -1 to get 2.

$$\frac{5^{-1}a^{2}\left(b^{8}\right)^{-1}}{\left(3c^{-2}\right)^{-1}}$$

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

$$\frac{5^{-1}a^{2}b^{-8}}{\left(3c^{-2}\right)^{-1}}$$

Calculate 5 to the power of -1 and get \frac{1}{5}.

$$\frac{\frac{1}{5}a^{2}b^{-8}}{\left(3c^{-2}\right)^{-1}}$$

Expand \left(3c^{-2}\right)^{-1}.

$$\frac{\frac{1}{5}a^{2}b^{-8}}{3^{-1}\left(c^{-2}\right)^{-1}}$$

To raise a power to another power, multiply the exponents. Multiply -2 and -1 to get 2.

$$\frac{\frac{1}{5}a^{2}b^{-8}}{3^{-1}c^{2}}$$

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

$$\frac{\frac{1}{5}a^{2}b^{-8}}{\frac{1}{3}c^{2}}$$

Differentiate w.r.t. a

$$\frac{6a}{5c^{2}b^{8}}$$