Solve (ab)^{3}*3a^2*(4a^2b^3)^2

$$( a b ) ^ { 3 } \cdot 3 a ^ { 2 } \cdot ( 4 a ^ { 2 } b ^ { 3 } ) ^ { 2 }$$

Evaluate

$$48\left(ab\right)^{9}$$

Solution Steps

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

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

To multiply powers of the same base, add their exponents. Add 3 and 2 to get 5.

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

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

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

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

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

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

$$a^{5}b^{3}\times 3\times 4^{2}a^{4}b^{6}$$

Calculate 4 to the power of 2 and get 16.

$$a^{5}b^{3}\times 3\times 16a^{4}b^{6}$$

Multiply 3 and 16 to get 48.

$$a^{5}b^{3}\times 48a^{4}b^{6}$$

To multiply powers of the same base, add their exponents. Add 5 and 4 to get 9.

$$a^{9}b^{3}\times 48b^{6}$$

To multiply powers of the same base, add their exponents. Add 3 and 6 to get 9.

$$a^{9}b^{9}\times 48$$

Expand

$$48\left(ab\right)^{9}$$

Solution Steps

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

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

To multiply powers of the same base, add their exponents. Add 3 and 2 to get 5.

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

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

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

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

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

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

$$a^{5}b^{3}\times 3\times 4^{2}a^{4}b^{6}$$

Calculate 4 to the power of 2 and get 16.

$$a^{5}b^{3}\times 3\times 16a^{4}b^{6}$$

Multiply 3 and 16 to get 48.

$$a^{5}b^{3}\times 48a^{4}b^{6}$$

To multiply powers of the same base, add their exponents. Add 5 and 4 to get 9.

$$a^{9}b^{3}\times 48b^{6}$$

To multiply powers of the same base, add their exponents. Add 3 and 6 to get 9.

$$a^{9}b^{9}\times 48$$