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

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

Evaluate

$$72b^{14}\left(ac\right)^{22}$$

Solution Steps

Expand \left(3a^{5}b^{4}c^{5}\right)^{2}.

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

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

$$3^{2}a^{10}\left(b^{4}\right)^{2}\left(c^{5}\right)^{2}\times \left(2a^{4}b^{2}c^{4}\right)^{3}$$

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

$$3^{2}a^{10}b^{8}\left(c^{5}\right)^{2}\times \left(2a^{4}b^{2}c^{4}\right)^{3}$$

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

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

Calculate 3 to the power of 2 and get 9.

$$9a^{10}b^{8}c^{10}\times \left(2a^{4}b^{2}c^{4}\right)^{3}$$

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

$$9a^{10}b^{8}c^{10}\times 2^{3}\left(a^{4}\right)^{3}\left(b^{2}\right)^{3}\left(c^{4}\right)^{3}$$

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

$$9a^{10}b^{8}c^{10}\times 2^{3}a^{12}\left(b^{2}\right)^{3}\left(c^{4}\right)^{3}$$

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

$$9a^{10}b^{8}c^{10}\times 2^{3}a^{12}b^{6}\left(c^{4}\right)^{3}$$

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

$$9a^{10}b^{8}c^{10}\times 2^{3}a^{12}b^{6}c^{12}$$

Calculate 2 to the power of 3 and get 8.

$$9a^{10}b^{8}c^{10}\times 8a^{12}b^{6}c^{12}$$

Multiply 9 and 8 to get 72.

$$72a^{10}b^{8}c^{10}a^{12}b^{6}c^{12}$$

To multiply powers of the same base, add their exponents. Add 10 and 12 to get 22.

$$72a^{22}b^{8}c^{10}b^{6}c^{12}$$

To multiply powers of the same base, add their exponents. Add 8 and 6 to get 14.

$$72a^{22}b^{14}c^{10}c^{12}$$

To multiply powers of the same base, add their exponents. Add 10 and 12 to get 22.

$$72a^{22}b^{14}c^{22}$$

Expand

$$72b^{14}\left(ac\right)^{22}$$

Solution Steps

Expand \left(3a^{5}b^{4}c^{5}\right)^{2}.

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

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

$$3^{2}a^{10}\left(b^{4}\right)^{2}\left(c^{5}\right)^{2}\times \left(2a^{4}b^{2}c^{4}\right)^{3}$$

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

$$3^{2}a^{10}b^{8}\left(c^{5}\right)^{2}\times \left(2a^{4}b^{2}c^{4}\right)^{3}$$

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

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

Calculate 3 to the power of 2 and get 9.

$$9a^{10}b^{8}c^{10}\times \left(2a^{4}b^{2}c^{4}\right)^{3}$$

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

$$9a^{10}b^{8}c^{10}\times 2^{3}\left(a^{4}\right)^{3}\left(b^{2}\right)^{3}\left(c^{4}\right)^{3}$$

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

$$9a^{10}b^{8}c^{10}\times 2^{3}a^{12}\left(b^{2}\right)^{3}\left(c^{4}\right)^{3}$$

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

$$9a^{10}b^{8}c^{10}\times 2^{3}a^{12}b^{6}\left(c^{4}\right)^{3}$$

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

$$9a^{10}b^{8}c^{10}\times 2^{3}a^{12}b^{6}c^{12}$$

Calculate 2 to the power of 3 and get 8.

$$9a^{10}b^{8}c^{10}\times 8a^{12}b^{6}c^{12}$$

Multiply 9 and 8 to get 72.

$$72a^{10}b^{8}c^{10}a^{12}b^{6}c^{12}$$

To multiply powers of the same base, add their exponents. Add 10 and 12 to get 22.

$$72a^{22}b^{8}c^{10}b^{6}c^{12}$$

To multiply powers of the same base, add their exponents. Add 8 and 6 to get 14.

$$72a^{22}b^{14}c^{10}c^{12}$$

To multiply powers of the same base, add their exponents. Add 10 and 12 to get 22.

$$72a^{22}b^{14}c^{22}$$