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

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

Evaluate

$$1000a^{12}b^{19}$$

Solution Steps

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

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

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

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

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

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

Calculate 2 to the power of 2 and get 4.

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

Multiply 2 and 4 to get 8.

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

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

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

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

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

To raise a power to another power, multiply the exponents. Multiply 5 and 3 to get 15.

$$8a^{6}b^{4}\times 5^{3}a^{6}b^{15}$$

Calculate 5 to the power of 3 and get 125.

$$8a^{6}b^{4}\times 125a^{6}b^{15}$$

Multiply 8 and 125 to get 1000.

$$1000a^{6}b^{4}a^{6}b^{15}$$

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

$$1000a^{12}b^{4}b^{15}$$

To multiply powers of the same base, add their exponents. Add 4 and 15 to get 19.

$$1000a^{12}b^{19}$$

Expand

$$1000a^{12}b^{19}$$

Solution Steps

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

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

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

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

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

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

Calculate 2 to the power of 2 and get 4.

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

Multiply 2 and 4 to get 8.

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

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

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

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

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

To raise a power to another power, multiply the exponents. Multiply 5 and 3 to get 15.

$$8a^{6}b^{4}\times 5^{3}a^{6}b^{15}$$

Calculate 5 to the power of 3 and get 125.

$$8a^{6}b^{4}\times 125a^{6}b^{15}$$

Multiply 8 and 125 to get 1000.

$$1000a^{6}b^{4}a^{6}b^{15}$$

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

$$1000a^{12}b^{4}b^{15}$$

To multiply powers of the same base, add their exponents. Add 4 and 15 to get 19.

$$1000a^{12}b^{19}$$