Solve (6ab^{2}c^{2})^2(-3ab^2c^4)^-1

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

Evaluate

$$-12ab^{2}$$

Solution Steps

Expand \left(6ab^{2}c^{2}\right)^{2}.

$$6^{2}a^{2}\left(b^{2}\right)^{2}\left(c^{2}\right)^{2}\left(-3ab^{2}c^{4}\right)^{-1}$$

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

$$6^{2}a^{2}b^{4}\left(c^{2}\right)^{2}\left(-3ab^{2}c^{4}\right)^{-1}$$

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

$$6^{2}a^{2}b^{4}c^{4}\left(-3ab^{2}c^{4}\right)^{-1}$$

Calculate 6 to the power of 2 and get 36.

$$36a^{2}b^{4}c^{4}\left(-3ab^{2}c^{4}\right)^{-1}$$

Expand \left(-3ab^{2}c^{4}\right)^{-1}.

$$36a^{2}b^{4}c^{4}\left(-3\right)^{-1}a^{-1}\left(b^{2}\right)^{-1}\left(c^{4}\right)^{-1}$$

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

$$36a^{2}b^{4}c^{4}\left(-3\right)^{-1}a^{-1}b^{-2}\left(c^{4}\right)^{-1}$$

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

$$36a^{2}b^{4}c^{4}\left(-3\right)^{-1}a^{-1}b^{-2}c^{-4}$$

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

$$36a^{2}b^{4}c^{4}\left(-\frac{1}{3}\right)a^{-1}b^{-2}c^{-4}$$

Multiply 36 and -\frac{1}{3} to get -12.

$$-12a^{2}b^{4}c^{4}a^{-1}b^{-2}c^{-4}$$

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

$$-12a^{1}b^{4}c^{4}b^{-2}c^{-4}$$

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

$$-12a^{1}b^{2}c^{4}c^{-4}$$

Multiply c^{4} and c^{-4} to get 1.

$$-12a^{1}b^{2}$$

Calculate a to the power of 1 and get a.

$$-12ab^{2}$$

Expand

$$-12ab^{2}$$

Solution Steps

Expand \left(6ab^{2}c^{2}\right)^{2}.

$$6^{2}a^{2}\left(b^{2}\right)^{2}\left(c^{2}\right)^{2}\left(-3ab^{2}c^{4}\right)^{-1}$$

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

$$6^{2}a^{2}b^{4}\left(c^{2}\right)^{2}\left(-3ab^{2}c^{4}\right)^{-1}$$

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

$$6^{2}a^{2}b^{4}c^{4}\left(-3ab^{2}c^{4}\right)^{-1}$$

Calculate 6 to the power of 2 and get 36.

$$36a^{2}b^{4}c^{4}\left(-3ab^{2}c^{4}\right)^{-1}$$

Expand \left(-3ab^{2}c^{4}\right)^{-1}.

$$36a^{2}b^{4}c^{4}\left(-3\right)^{-1}a^{-1}\left(b^{2}\right)^{-1}\left(c^{4}\right)^{-1}$$

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

$$36a^{2}b^{4}c^{4}\left(-3\right)^{-1}a^{-1}b^{-2}\left(c^{4}\right)^{-1}$$

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

$$36a^{2}b^{4}c^{4}\left(-3\right)^{-1}a^{-1}b^{-2}c^{-4}$$

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

$$36a^{2}b^{4}c^{4}\left(-\frac{1}{3}\right)a^{-1}b^{-2}c^{-4}$$

Multiply 36 and -\frac{1}{3} to get -12.

$$-12a^{2}b^{4}c^{4}a^{-1}b^{-2}c^{-4}$$

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

$$-12a^{1}b^{4}c^{4}b^{-2}c^{-4}$$

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

$$-12a^{1}b^{2}c^{4}c^{-4}$$

Multiply c^{4} and c^{-4} to get 1.

$$-12a^{1}b^{2}$$

Calculate a to the power of 1 and get a.

$$-12ab^{2}$$