Solve (2mn^{2})^-2(m^-2n^-1)^-3

$$( 2 m n ^ { 2 } ) ^ { - 2 } ( m ^ { - 2 } n ^ { - 1 } ) ^ { - 3 }$$

Evaluate

$$\frac{m^{4}}{4n}$$

Solution Steps

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

$$2^{-2}m^{-2}\left(n^{2}\right)^{-2}\left(m^{-2}n^{-1}\right)^{-3}$$

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

$$2^{-2}m^{-2}n^{-4}\left(m^{-2}n^{-1}\right)^{-3}$$

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

$$\frac{1}{4}m^{-2}n^{-4}\left(m^{-2}n^{-1}\right)^{-3}$$

Expand \left(m^{-2}n^{-1}\right)^{-3}.

$$\frac{1}{4}m^{-2}n^{-4}\left(m^{-2}\right)^{-3}\left(n^{-1}\right)^{-3}$$

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

$$\frac{1}{4}m^{-2}n^{-4}m^{6}\left(n^{-1}\right)^{-3}$$

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

$$\frac{1}{4}m^{-2}n^{-4}m^{6}n^{3}$$

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

$$\frac{1}{4}m^{4}n^{-4}n^{3}$$

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

$$\frac{1}{4}m^{4}n^{-1}$$

Expand

$$\frac{m^{4}}{4n}$$

Solution Steps

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

$$2^{-2}m^{-2}\left(n^{2}\right)^{-2}\left(m^{-2}n^{-1}\right)^{-3}$$

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

$$2^{-2}m^{-2}n^{-4}\left(m^{-2}n^{-1}\right)^{-3}$$

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

$$\frac{1}{4}m^{-2}n^{-4}\left(m^{-2}n^{-1}\right)^{-3}$$

Expand \left(m^{-2}n^{-1}\right)^{-3}.

$$\frac{1}{4}m^{-2}n^{-4}\left(m^{-2}\right)^{-3}\left(n^{-1}\right)^{-3}$$

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

$$\frac{1}{4}m^{-2}n^{-4}m^{6}\left(n^{-1}\right)^{-3}$$

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

$$\frac{1}{4}m^{-2}n^{-4}m^{6}n^{3}$$

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

$$\frac{1}{4}m^{4}n^{-4}n^{3}$$

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

$$\frac{1}{4}m^{4}n^{-1}$$