Solve (3ab^{2}c)^{2}(-2a^{2}b^{4})^{2}(a^{4}c^{2})^{3}(a^{2}b^{4}c^{5})^{2}(2a^3b^2c^4)^3

$$( 3 a b ^ { 2 } c ) ^ { 2 } ( - 2 a ^ { 2 } b ^ { 4 } ) ^ { 2 } ( a ^ { 4 } c ^ { 2 } ) ^ { 3 } ( a ^ { 2 } b ^ { 4 } c ^ { 5 } ) ^ { 2 } ( 2 a ^ { 3 } b ^ { 2 } c ^ { 4 } ) ^ { 3 }$$

Evaluate

$$288b^{26}c^{30}a^{31}$$

Solution Steps

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

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

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

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

Calculate 3 to the power of 2 and get 9.

$$9a^{2}b^{4}c^{2}\left(-2a^{2}b^{4}\right)^{2}\left(a^{4}c^{2}\right)^{3}\left(a^{2}b^{4}c^{5}\right)^{2}\times \left(2a^{3}b^{2}c^{4}\right)^{3}$$

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

$$9a^{2}b^{4}c^{2}\left(-2\right)^{2}\left(a^{2}\right)^{2}\left(b^{4}\right)^{2}\left(a^{4}c^{2}\right)^{3}\left(a^{2}b^{4}c^{5}\right)^{2}\times \left(2a^{3}b^{2}c^{4}\right)^{3}$$

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

$$9a^{2}b^{4}c^{2}\left(-2\right)^{2}a^{4}\left(b^{4}\right)^{2}\left(a^{4}c^{2}\right)^{3}\left(a^{2}b^{4}c^{5}\right)^{2}\times \left(2a^{3}b^{2}c^{4}\right)^{3}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 2 to get 8.

$$9a^{2}b^{4}c^{2}\left(-2\right)^{2}a^{4}b^{8}\left(a^{4}c^{2}\right)^{3}\left(a^{2}b^{4}c^{5}\right)^{2}\times \left(2a^{3}b^{2}c^{4}\right)^{3}$$

Calculate -2 to the power of 2 and get 4.

$$9a^{2}b^{4}c^{2}\times 4a^{4}b^{8}\left(a^{4}c^{2}\right)^{3}\left(a^{2}b^{4}c^{5}\right)^{2}\times \left(2a^{3}b^{2}c^{4}\right)^{3}$$

Multiply 9 and 4 to get 36.

$$36a^{2}b^{4}c^{2}a^{4}b^{8}\left(a^{4}c^{2}\right)^{3}\left(a^{2}b^{4}c^{5}\right)^{2}\times \left(2a^{3}b^{2}c^{4}\right)^{3}$$

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

$$36a^{6}b^{4}c^{2}b^{8}\left(a^{4}c^{2}\right)^{3}\left(a^{2}b^{4}c^{5}\right)^{2}\times \left(2a^{3}b^{2}c^{4}\right)^{3}$$

To multiply powers of the same base, add their exponents. Add 4 and 8 to get 12.

$$36a^{6}b^{12}c^{2}\left(a^{4}c^{2}\right)^{3}\left(a^{2}b^{4}c^{5}\right)^{2}\times \left(2a^{3}b^{2}c^{4}\right)^{3}$$

Expand \left(a^{4}c^{2}\right)^{3}.

$$36a^{6}b^{12}c^{2}\left(a^{4}\right)^{3}\left(c^{2}\right)^{3}\left(a^{2}b^{4}c^{5}\right)^{2}\times \left(2a^{3}b^{2}c^{4}\right)^{3}$$

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

$$36a^{6}b^{12}c^{2}a^{12}\left(c^{2}\right)^{3}\left(a^{2}b^{4}c^{5}\right)^{2}\times \left(2a^{3}b^{2}c^{4}\right)^{3}$$

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

$$36a^{6}b^{12}c^{2}a^{12}c^{6}\left(a^{2}b^{4}c^{5}\right)^{2}\times \left(2a^{3}b^{2}c^{4}\right)^{3}$$

To multiply powers of the same base, add their exponents. Add 6 and 12 to get 18.

$$36a^{18}b^{12}c^{2}c^{6}\left(a^{2}b^{4}c^{5}\right)^{2}\times \left(2a^{3}b^{2}c^{4}\right)^{3}$$

To multiply powers of the same base, add their exponents. Add 2 and 6 to get 8.

$$36a^{18}b^{12}c^{8}\left(a^{2}b^{4}c^{5}\right)^{2}\times \left(2a^{3}b^{2}c^{4}\right)^{3}$$

Expand \left(a^{2}b^{4}c^{5}\right)^{2}.

$$36a^{18}b^{12}c^{8}\left(a^{2}\right)^{2}\left(b^{4}\right)^{2}\left(c^{5}\right)^{2}\times \left(2a^{3}b^{2}c^{4}\right)^{3}$$

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

$$36a^{18}b^{12}c^{8}a^{4}\left(b^{4}\right)^{2}\left(c^{5}\right)^{2}\times \left(2a^{3}b^{2}c^{4}\right)^{3}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 2 to get 8.

$$36a^{18}b^{12}c^{8}a^{4}b^{8}\left(c^{5}\right)^{2}\times \left(2a^{3}b^{2}c^{4}\right)^{3}$$

To raise a power to another power, multiply the exponents. Multiply 5 and 2 to get 10.

$$36a^{18}b^{12}c^{8}a^{4}b^{8}c^{10}\times \left(2a^{3}b^{2}c^{4}\right)^{3}$$

To multiply powers of the same base, add their exponents. Add 18 and 4 to get 22.

$$36a^{22}b^{12}c^{8}b^{8}c^{10}\times \left(2a^{3}b^{2}c^{4}\right)^{3}$$

To multiply powers of the same base, add their exponents. Add 12 and 8 to get 20.

$$36a^{22}b^{20}c^{8}c^{10}\times \left(2a^{3}b^{2}c^{4}\right)^{3}$$

To multiply powers of the same base, add their exponents. Add 8 and 10 to get 18.

$$36a^{22}b^{20}c^{18}\times \left(2a^{3}b^{2}c^{4}\right)^{3}$$

Expand \left(2a^{3}b^{2}c^{4}\right)^{3}.

$$36a^{22}b^{20}c^{18}\times 2^{3}\left(a^{3}\right)^{3}\left(b^{2}\right)^{3}\left(c^{4}\right)^{3}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 3 to get 9.

$$36a^{22}b^{20}c^{18}\times 2^{3}a^{9}\left(b^{2}\right)^{3}\left(c^{4}\right)^{3}$$

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

$$36a^{22}b^{20}c^{18}\times 2^{3}a^{9}b^{6}\left(c^{4}\right)^{3}$$

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

$$36a^{22}b^{20}c^{18}\times 2^{3}a^{9}b^{6}c^{12}$$

Calculate 2 to the power of 3 and get 8.

$$36a^{22}b^{20}c^{18}\times 8a^{9}b^{6}c^{12}$$

Multiply 36 and 8 to get 288.

$$288a^{22}b^{20}c^{18}a^{9}b^{6}c^{12}$$

To multiply powers of the same base, add their exponents. Add 22 and 9 to get 31.

$$288a^{31}b^{20}c^{18}b^{6}c^{12}$$

To multiply powers of the same base, add their exponents. Add 20 and 6 to get 26.

$$288a^{31}b^{26}c^{18}c^{12}$$

To multiply powers of the same base, add their exponents. Add 18 and 12 to get 30.

$$288a^{31}b^{26}c^{30}$$

Expand

$$288b^{26}c^{30}a^{31}$$

Solution Steps

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

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

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

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

Calculate 3 to the power of 2 and get 9.

$$9a^{2}b^{4}c^{2}\left(-2a^{2}b^{4}\right)^{2}\left(a^{4}c^{2}\right)^{3}\left(a^{2}b^{4}c^{5}\right)^{2}\times \left(2a^{3}b^{2}c^{4}\right)^{3}$$

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

$$9a^{2}b^{4}c^{2}\left(-2\right)^{2}\left(a^{2}\right)^{2}\left(b^{4}\right)^{2}\left(a^{4}c^{2}\right)^{3}\left(a^{2}b^{4}c^{5}\right)^{2}\times \left(2a^{3}b^{2}c^{4}\right)^{3}$$

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

$$9a^{2}b^{4}c^{2}\left(-2\right)^{2}a^{4}\left(b^{4}\right)^{2}\left(a^{4}c^{2}\right)^{3}\left(a^{2}b^{4}c^{5}\right)^{2}\times \left(2a^{3}b^{2}c^{4}\right)^{3}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 2 to get 8.

$$9a^{2}b^{4}c^{2}\left(-2\right)^{2}a^{4}b^{8}\left(a^{4}c^{2}\right)^{3}\left(a^{2}b^{4}c^{5}\right)^{2}\times \left(2a^{3}b^{2}c^{4}\right)^{3}$$

Calculate -2 to the power of 2 and get 4.

$$9a^{2}b^{4}c^{2}\times 4a^{4}b^{8}\left(a^{4}c^{2}\right)^{3}\left(a^{2}b^{4}c^{5}\right)^{2}\times \left(2a^{3}b^{2}c^{4}\right)^{3}$$

Multiply 9 and 4 to get 36.

$$36a^{2}b^{4}c^{2}a^{4}b^{8}\left(a^{4}c^{2}\right)^{3}\left(a^{2}b^{4}c^{5}\right)^{2}\times \left(2a^{3}b^{2}c^{4}\right)^{3}$$

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

$$36a^{6}b^{4}c^{2}b^{8}\left(a^{4}c^{2}\right)^{3}\left(a^{2}b^{4}c^{5}\right)^{2}\times \left(2a^{3}b^{2}c^{4}\right)^{3}$$

To multiply powers of the same base, add their exponents. Add 4 and 8 to get 12.

$$36a^{6}b^{12}c^{2}\left(a^{4}c^{2}\right)^{3}\left(a^{2}b^{4}c^{5}\right)^{2}\times \left(2a^{3}b^{2}c^{4}\right)^{3}$$

Expand \left(a^{4}c^{2}\right)^{3}.

$$36a^{6}b^{12}c^{2}\left(a^{4}\right)^{3}\left(c^{2}\right)^{3}\left(a^{2}b^{4}c^{5}\right)^{2}\times \left(2a^{3}b^{2}c^{4}\right)^{3}$$

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

$$36a^{6}b^{12}c^{2}a^{12}\left(c^{2}\right)^{3}\left(a^{2}b^{4}c^{5}\right)^{2}\times \left(2a^{3}b^{2}c^{4}\right)^{3}$$

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

$$36a^{6}b^{12}c^{2}a^{12}c^{6}\left(a^{2}b^{4}c^{5}\right)^{2}\times \left(2a^{3}b^{2}c^{4}\right)^{3}$$

To multiply powers of the same base, add their exponents. Add 6 and 12 to get 18.

$$36a^{18}b^{12}c^{2}c^{6}\left(a^{2}b^{4}c^{5}\right)^{2}\times \left(2a^{3}b^{2}c^{4}\right)^{3}$$

To multiply powers of the same base, add their exponents. Add 2 and 6 to get 8.

$$36a^{18}b^{12}c^{8}\left(a^{2}b^{4}c^{5}\right)^{2}\times \left(2a^{3}b^{2}c^{4}\right)^{3}$$

Expand \left(a^{2}b^{4}c^{5}\right)^{2}.

$$36a^{18}b^{12}c^{8}\left(a^{2}\right)^{2}\left(b^{4}\right)^{2}\left(c^{5}\right)^{2}\times \left(2a^{3}b^{2}c^{4}\right)^{3}$$

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

$$36a^{18}b^{12}c^{8}a^{4}\left(b^{4}\right)^{2}\left(c^{5}\right)^{2}\times \left(2a^{3}b^{2}c^{4}\right)^{3}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 2 to get 8.

$$36a^{18}b^{12}c^{8}a^{4}b^{8}\left(c^{5}\right)^{2}\times \left(2a^{3}b^{2}c^{4}\right)^{3}$$

To raise a power to another power, multiply the exponents. Multiply 5 and 2 to get 10.

$$36a^{18}b^{12}c^{8}a^{4}b^{8}c^{10}\times \left(2a^{3}b^{2}c^{4}\right)^{3}$$

To multiply powers of the same base, add their exponents. Add 18 and 4 to get 22.

$$36a^{22}b^{12}c^{8}b^{8}c^{10}\times \left(2a^{3}b^{2}c^{4}\right)^{3}$$

To multiply powers of the same base, add their exponents. Add 12 and 8 to get 20.

$$36a^{22}b^{20}c^{8}c^{10}\times \left(2a^{3}b^{2}c^{4}\right)^{3}$$

To multiply powers of the same base, add their exponents. Add 8 and 10 to get 18.

$$36a^{22}b^{20}c^{18}\times \left(2a^{3}b^{2}c^{4}\right)^{3}$$

Expand \left(2a^{3}b^{2}c^{4}\right)^{3}.

$$36a^{22}b^{20}c^{18}\times 2^{3}\left(a^{3}\right)^{3}\left(b^{2}\right)^{3}\left(c^{4}\right)^{3}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 3 to get 9.

$$36a^{22}b^{20}c^{18}\times 2^{3}a^{9}\left(b^{2}\right)^{3}\left(c^{4}\right)^{3}$$

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

$$36a^{22}b^{20}c^{18}\times 2^{3}a^{9}b^{6}\left(c^{4}\right)^{3}$$

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

$$36a^{22}b^{20}c^{18}\times 2^{3}a^{9}b^{6}c^{12}$$

Calculate 2 to the power of 3 and get 8.

$$36a^{22}b^{20}c^{18}\times 8a^{9}b^{6}c^{12}$$

Multiply 36 and 8 to get 288.

$$288a^{22}b^{20}c^{18}a^{9}b^{6}c^{12}$$

To multiply powers of the same base, add their exponents. Add 22 and 9 to get 31.

$$288a^{31}b^{20}c^{18}b^{6}c^{12}$$

To multiply powers of the same base, add their exponents. Add 20 and 6 to get 26.

$$288a^{31}b^{26}c^{18}c^{12}$$

To multiply powers of the same base, add their exponents. Add 18 and 12 to get 30.

$$288a^{31}b^{26}c^{30}$$