Solve (3a^{2}b^{3})^2(2a^-1b^-2)^-1

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

Evaluate

$$\frac{9a^{5}b^{8}}{2}$$

Solution Steps

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

$$3^{2}\left(a^{2}\right)^{2}\left(b^{3}\right)^{2}\times \left(2a^{-1}b^{-2}\right)^{-1}$$

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

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

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

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

Calculate 3 to the power of 2 and get 9.

$$9a^{4}b^{6}\times \left(2a^{-1}b^{-2}\right)^{-1}$$

Expand \left(2a^{-1}b^{-2}\right)^{-1}.

$$9a^{4}b^{6}\times 2^{-1}\left(a^{-1}\right)^{-1}\left(b^{-2}\right)^{-1}$$

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

$$9a^{4}b^{6}\times 2^{-1}a^{1}\left(b^{-2}\right)^{-1}$$

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

$$9a^{4}b^{6}\times 2^{-1}a^{1}b^{2}$$

Calculate 2 to the power of -1 and get \frac{1}{2}.

$$9a^{4}b^{6}\times \frac{1}{2}a^{1}b^{2}$$

Calculate a to the power of 1 and get a.

$$9a^{4}b^{6}\times \frac{1}{2}ab^{2}$$

Multiply 9 and \frac{1}{2} to get \frac{9}{2}.

$$\frac{9}{2}a^{4}b^{6}ab^{2}$$

To multiply powers of the same base, add their exponents. Add 4 and 1 to get 5.

$$\frac{9}{2}a^{5}b^{6}b^{2}$$

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

$$\frac{9}{2}a^{5}b^{8}$$

Expand

$$\frac{9a^{5}b^{8}}{2}$$

Solution Steps

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

$$3^{2}\left(a^{2}\right)^{2}\left(b^{3}\right)^{2}\times \left(2a^{-1}b^{-2}\right)^{-1}$$

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

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

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

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

Calculate 3 to the power of 2 and get 9.

$$9a^{4}b^{6}\times \left(2a^{-1}b^{-2}\right)^{-1}$$

Expand \left(2a^{-1}b^{-2}\right)^{-1}.

$$9a^{4}b^{6}\times 2^{-1}\left(a^{-1}\right)^{-1}\left(b^{-2}\right)^{-1}$$

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

$$9a^{4}b^{6}\times 2^{-1}a^{1}\left(b^{-2}\right)^{-1}$$

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

$$9a^{4}b^{6}\times 2^{-1}a^{1}b^{2}$$

Calculate 2 to the power of -1 and get \frac{1}{2}.

$$9a^{4}b^{6}\times \frac{1}{2}a^{1}b^{2}$$

Calculate a to the power of 1 and get a.

$$9a^{4}b^{6}\times \frac{1}{2}ab^{2}$$

Multiply 9 and \frac{1}{2} to get \frac{9}{2}.

$$\frac{9}{2}a^{4}b^{6}ab^{2}$$

To multiply powers of the same base, add their exponents. Add 4 and 1 to get 5.

$$\frac{9}{2}a^{5}b^{6}b^{2}$$

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

$$\frac{9}{2}a^{5}b^{8}$$