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

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

Evaluate

$$\frac{a^{4}}{4b}$$

Solution Steps

Expand \left(a^{-2}b^{-1}\right)^{-3}.

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

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

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

To raise a power to another power, multiply the exponents. Multiply -1 and -3 to get 3.

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

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

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

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

$$1a^{6}b^{3}\times 2^{-2}a^{-2}b^{-4}$$

Calculate 2 to the power of -2 and get \frac{1}{4}.

$$1a^{6}b^{3}\times \frac{1}{4}a^{-2}b^{-4}$$

Multiply 1 and \frac{1}{4} to get \frac{1}{4}.

$$\frac{1}{4}a^{6}b^{3}a^{-2}b^{-4}$$

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

$$\frac{1}{4}a^{4}b^{3}b^{-4}$$

To multiply powers of the same base, add their exponents. Add 3 and -4 to get -1.

$$\frac{1}{4}a^{4}b^{-1}$$

Expand

$$\frac{a^{4}}{4b}$$

Solution Steps

Expand \left(a^{-2}b^{-1}\right)^{-3}.

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

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

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

To raise a power to another power, multiply the exponents. Multiply -1 and -3 to get 3.

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

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

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

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

$$1a^{6}b^{3}\times 2^{-2}a^{-2}b^{-4}$$

Calculate 2 to the power of -2 and get \frac{1}{4}.

$$1a^{6}b^{3}\times \frac{1}{4}a^{-2}b^{-4}$$

Multiply 1 and \frac{1}{4} to get \frac{1}{4}.

$$\frac{1}{4}a^{6}b^{3}a^{-2}b^{-4}$$

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

$$\frac{1}{4}a^{4}b^{3}b^{-4}$$

To multiply powers of the same base, add their exponents. Add 3 and -4 to get -1.

$$\frac{1}{4}a^{4}b^{-1}$$