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

$$( a ^ { - 1 } b ^ { - 4 } c ) ^ { - 3 } ( a ^ { 2 } b ^ { - 1 } c ^ { 3 } ) ^ { 2 }$$

Evaluate

$$c^{3}a^{7}b^{10}$$

Solution Steps

Expand \left(a^{-1}b^{-4}c\right)^{-3}.

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

To raise a power to another power, multiply the exponents. Multiply -1 and -3 to get 3.

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

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

$$a^{3}b^{12}c^{-3}\left(a^{2}b^{-1}c^{3}\right)^{2}$$

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

$$a^{3}b^{12}c^{-3}\left(a^{2}\right)^{2}\left(b^{-1}\right)^{2}\left(c^{3}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$a^{3}b^{12}c^{-3}a^{4}\left(b^{-1}\right)^{2}\left(c^{3}\right)^{2}$$

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

$$a^{3}b^{12}c^{-3}a^{4}b^{-2}\left(c^{3}\right)^{2}$$

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

$$a^{3}b^{12}c^{-3}a^{4}b^{-2}c^{6}$$

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

$$a^{7}b^{12}c^{-3}b^{-2}c^{6}$$

To multiply powers of the same base, add their exponents. Add 12 and -2 to get 10.

$$a^{7}b^{10}c^{-3}c^{6}$$

To multiply powers of the same base, add their exponents. Add -3 and 6 to get 3.

$$a^{7}b^{10}c^{3}$$

Expand

$$c^{3}a^{7}b^{10}$$

Solution Steps

Expand \left(a^{-1}b^{-4}c\right)^{-3}.

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

To raise a power to another power, multiply the exponents. Multiply -1 and -3 to get 3.

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

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

$$a^{3}b^{12}c^{-3}\left(a^{2}b^{-1}c^{3}\right)^{2}$$

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

$$a^{3}b^{12}c^{-3}\left(a^{2}\right)^{2}\left(b^{-1}\right)^{2}\left(c^{3}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$a^{3}b^{12}c^{-3}a^{4}\left(b^{-1}\right)^{2}\left(c^{3}\right)^{2}$$

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

$$a^{3}b^{12}c^{-3}a^{4}b^{-2}\left(c^{3}\right)^{2}$$

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

$$a^{3}b^{12}c^{-3}a^{4}b^{-2}c^{6}$$

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

$$a^{7}b^{12}c^{-3}b^{-2}c^{6}$$

To multiply powers of the same base, add their exponents. Add 12 and -2 to get 10.

$$a^{7}b^{10}c^{-3}c^{6}$$

To multiply powers of the same base, add their exponents. Add -3 and 6 to get 3.

$$a^{7}b^{10}c^{3}$$