Solve frac{(0.5m^{-3}n^{2})^{-2}*(m^{2})^-4}{(4m^-2n^-3)^2}=

$$\frac { ( 0.5 m ^ { - 3 } n ^ { 2 } ) ^ { - 2 } \cdot ( m ^ { 2 } ) ^ { - 4 } } { ( 4 m ^ { - 2 } n ^ { - 3 } ) ^ { 2 } } =$$

Evaluate

$$\frac{\left(mn\right)^{2}}{4}$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply 2 and -4 to get -8.

$$\frac{\left(0.5m^{-3}n^{2}\right)^{-2}m^{-8}}{\left(4m^{-2}n^{-3}\right)^{2}}$$

Expand \left(0.5m^{-3}n^{2}\right)^{-2}.

$$\frac{0.5^{-2}\left(m^{-3}\right)^{-2}\left(n^{2}\right)^{-2}m^{-8}}{\left(4m^{-2}n^{-3}\right)^{2}}$$

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

$$\frac{0.5^{-2}m^{6}\left(n^{2}\right)^{-2}m^{-8}}{\left(4m^{-2}n^{-3}\right)^{2}}$$

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

$$\frac{0.5^{-2}m^{6}n^{-4}m^{-8}}{\left(4m^{-2}n^{-3}\right)^{2}}$$

Calculate 0.5 to the power of -2 and get 4.

$$\frac{4m^{6}n^{-4}m^{-8}}{\left(4m^{-2}n^{-3}\right)^{2}}$$

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

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

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

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

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

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

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

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

Calculate 4 to the power of 2 and get 16.

$$\frac{4m^{-2}n^{-4}}{16m^{-4}n^{-6}}$$

Cancel out 4 in both numerator and denominator.

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

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\frac{m^{2}n^{2}}{4}$$

Expand

$$\frac{\left(mn\right)^{2}}{4}$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply 2 and -4 to get -8.

$$\frac{\left(0.5m^{-3}n^{2}\right)^{-2}m^{-8}}{\left(4m^{-2}n^{-3}\right)^{2}}$$

Expand \left(0.5m^{-3}n^{2}\right)^{-2}.

$$\frac{0.5^{-2}\left(m^{-3}\right)^{-2}\left(n^{2}\right)^{-2}m^{-8}}{\left(4m^{-2}n^{-3}\right)^{2}}$$

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

$$\frac{0.5^{-2}m^{6}\left(n^{2}\right)^{-2}m^{-8}}{\left(4m^{-2}n^{-3}\right)^{2}}$$

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

$$\frac{0.5^{-2}m^{6}n^{-4}m^{-8}}{\left(4m^{-2}n^{-3}\right)^{2}}$$

Calculate 0.5 to the power of -2 and get 4.

$$\frac{4m^{6}n^{-4}m^{-8}}{\left(4m^{-2}n^{-3}\right)^{2}}$$

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

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

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

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

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

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

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

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

Calculate 4 to the power of 2 and get 16.

$$\frac{4m^{-2}n^{-4}}{16m^{-4}n^{-6}}$$

Cancel out 4 in both numerator and denominator.

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

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\frac{m^{2}n^{2}}{4}$$