Solve (-6mn^{-2})^3(-4m^-3n^-1)^-2

$$( - 6 m n ^ { - 2 } ) ^ { 3 } ( - 4 m ^ { - 3 } n ^ { - 1 } ) ^ { - 2 }$$

Evaluate

$$-\frac{27m^{9}}{2n^{4}}$$

Solution Steps

Expand \left(-6mn^{-2}\right)^{3}.

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

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

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

Calculate -6 to the power of 3 and get -216.

$$-216m^{3}n^{-6}\left(-4m^{-3}n^{-1}\right)^{-2}$$

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

$$-216m^{3}n^{-6}\left(-4\right)^{-2}\left(m^{-3}\right)^{-2}\left(n^{-1}\right)^{-2}$$

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

$$-216m^{3}n^{-6}\left(-4\right)^{-2}m^{6}\left(n^{-1}\right)^{-2}$$

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

$$-216m^{3}n^{-6}\left(-4\right)^{-2}m^{6}n^{2}$$

Calculate -4 to the power of -2 and get \frac{1}{16}.

$$-216m^{3}n^{-6}\times \frac{1}{16}m^{6}n^{2}$$

Multiply -216 and \frac{1}{16} to get -\frac{27}{2}.

$$-\frac{27}{2}m^{3}n^{-6}m^{6}n^{2}$$

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

$$-\frac{27}{2}m^{9}n^{-6}n^{2}$$

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

$$-\frac{27}{2}m^{9}n^{-4}$$

Expand

$$-\frac{27m^{9}}{2n^{4}}$$

Solution Steps

Expand \left(-6mn^{-2}\right)^{3}.

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

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

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

Calculate -6 to the power of 3 and get -216.

$$-216m^{3}n^{-6}\left(-4m^{-3}n^{-1}\right)^{-2}$$

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

$$-216m^{3}n^{-6}\left(-4\right)^{-2}\left(m^{-3}\right)^{-2}\left(n^{-1}\right)^{-2}$$

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

$$-216m^{3}n^{-6}\left(-4\right)^{-2}m^{6}\left(n^{-1}\right)^{-2}$$

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

$$-216m^{3}n^{-6}\left(-4\right)^{-2}m^{6}n^{2}$$

Calculate -4 to the power of -2 and get \frac{1}{16}.

$$-216m^{3}n^{-6}\times \frac{1}{16}m^{6}n^{2}$$

Multiply -216 and \frac{1}{16} to get -\frac{27}{2}.

$$-\frac{27}{2}m^{3}n^{-6}m^{6}n^{2}$$

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

$$-\frac{27}{2}m^{9}n^{-6}n^{2}$$

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

$$-\frac{27}{2}m^{9}n^{-4}$$