Solve (a^{2}b^{-2}c)(a^-4b^2c^-2)^-1

$$( a ^ { 2 } b ^ { - 2 } c ) ( a ^ { - 4 } b ^ { 2 } c ^ { - 2 } ) ^ { - 1 }$$

Evaluate

$$\frac{c^{3}a^{6}}{b^{4}}$$

Solution Steps

Expand \left(a^{-4}b^{2}c^{-2}\right)^{-1}.

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

To raise a power to another power, multiply the exponents. Multiply -4 and -1 to get 4.

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

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

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

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

$$a^{2}b^{-2}ca^{4}b^{-2}c^{2}$$

To multiply powers of the same base, add their exponents. Add 2 and 4 to get 6.

$$a^{6}b^{-2}cb^{-2}c^{2}$$

To multiply powers of the same base, add their exponents. Add -2 and -2 to get -4.

$$a^{6}b^{-4}cc^{2}$$

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

$$a^{6}b^{-4}c^{3}$$

Differentiate w.r.t. a

$$\frac{6c^{3}a^{5}}{b^{4}}$$