Solve -2(m^2+2/2m^2)^2+4/m^2

$$- 2 ( \frac { m ^ { 2 } + 2 } { 2 m ^ { 2 } } ) ^ { 2 } + \frac { 4 } { m ^ { 2 } }$$

Evaluate

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

Short Solution Steps

To raise \frac{m^{2}+2}{2m^{2}} to a power, raise both numerator and denominator to the power and then divide.

$$-2\times \frac{\left(m^{2}+2\right)^{2}}{\left(2m^{2}\right)^{2}}+\frac{4}{m^{2}}$$

Express -2\times \frac{\left(m^{2}+2\right)^{2}}{\left(2m^{2}\right)^{2}} as a single fraction.

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

Expand \left(2m^{2}\right)^{2}.

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

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

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

Calculate 2 to the power of 2 and get 4.

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

Cancel out 2 in both numerator and denominator.

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

Cancel out -1 in both numerator and denominator.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of -2m^{4} and m^{2} is 2m^{4}. Multiply \frac{\left(m^{2}+2\right)^{2}}{-2m^{4}} times \frac{-1}{-1}. Multiply \frac{4}{m^{2}} times \frac{2m^{2}}{2m^{2}}.

$$\frac{-\left(m^{2}+2\right)^{2}}{2m^{4}}+\frac{4\times 2m^{2}}{2m^{4}}$$

Since \frac{-\left(m^{2}+2\right)^{2}}{2m^{4}} and \frac{4\times 2m^{2}}{2m^{4}} have the same denominator, add them by adding their numerators.

$$\frac{-\left(m^{2}+2\right)^{2}+4\times 2m^{2}}{2m^{4}}$$

Do the multiplications in -\left(m^{2}+2\right)^{2}+4\times 2m^{2}.

$$\frac{-m^{4}-4m^{2}-4+8m^{2}}{2m^{4}}$$

Combine like terms in -m^{4}-4m^{2}-4+8m^{2}.

$$\frac{-m^{4}+4m^{2}-4}{2m^{4}}$$

Expand

$$-\frac{1}{2}+\frac{2}{m^{2}}-\frac{2}{m^{4}}$$

Short Solution Steps

To raise \frac{m^{2}+2}{2m^{2}} to a power, raise both numerator and denominator to the power and then divide.

$$-2\times \frac{\left(m^{2}+2\right)^{2}}{\left(2m^{2}\right)^{2}}+\frac{4}{m^{2}}$$

Express -2\times \frac{\left(m^{2}+2\right)^{2}}{\left(2m^{2}\right)^{2}} as a single fraction.

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

Expand \left(2m^{2}\right)^{2}.

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

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

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

Calculate 2 to the power of 2 and get 4.

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

Cancel out 2 in both numerator and denominator.

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

Cancel out -1 in both numerator and denominator.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of -2m^{4} and m^{2} is 2m^{4}. Multiply \frac{\left(m^{2}+2\right)^{2}}{-2m^{4}} times \frac{-1}{-1}. Multiply \frac{4}{m^{2}} times \frac{2m^{2}}{2m^{2}}.

$$\frac{-\left(m^{2}+2\right)^{2}}{2m^{4}}+\frac{4\times 2m^{2}}{2m^{4}}$$

Since \frac{-\left(m^{2}+2\right)^{2}}{2m^{4}} and \frac{4\times 2m^{2}}{2m^{4}} have the same denominator, add them by adding their numerators.

$$\frac{-\left(m^{2}+2\right)^{2}+4\times 2m^{2}}{2m^{4}}$$

Do the multiplications in -\left(m^{2}+2\right)^{2}+4\times 2m^{2}.

$$\frac{-m^{4}-4m^{2}-4+8m^{2}}{2m^{4}}$$

Combine like terms in -m^{4}-4m^{2}-4+8m^{2}.

$$\frac{-m^{4}+4m^{2}-4}{2m^{4}}$$