Solve frac{m^{4}-49}{m^2+7}-m^2-343/m^4+7m^2+49

$$\frac { m ^ { 4 } - 49 } { m ^ { 2 } + 7 } - \frac { m ^ { 2 } - 343 } { m ^ { 4 } + 7 m ^ { 2 } + 49 }$$

Evaluate

$$\frac{m^{2}\left(m^{4}-1\right)}{m^{4}+7m^{2}+49}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{m^{4}-49}{m^{2}+7}.

$$\frac{\left(m^{2}-7\right)\left(m^{2}+7\right)}{m^{2}+7}-\frac{m^{2}-343}{m^{4}+7m^{2}+49}$$

Cancel out m^{2}+7 in both numerator and denominator.

$$m^{2}-7-\frac{m^{2}-343}{m^{4}+7m^{2}+49}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply m^{2}-7 times \frac{m^{4}+7m^{2}+49}{m^{4}+7m^{2}+49}.

$$\frac{\left(m^{2}-7\right)\left(m^{4}+7m^{2}+49\right)}{m^{4}+7m^{2}+49}-\frac{m^{2}-343}{m^{4}+7m^{2}+49}$$

Since \frac{\left(m^{2}-7\right)\left(m^{4}+7m^{2}+49\right)}{m^{4}+7m^{2}+49} and \frac{m^{2}-343}{m^{4}+7m^{2}+49} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(m^{2}-7\right)\left(m^{4}+7m^{2}+49\right)-\left(m^{2}-343\right)}{m^{4}+7m^{2}+49}$$

Do the multiplications in \left(m^{2}-7\right)\left(m^{4}+7m^{2}+49\right)-\left(m^{2}-343\right).

$$\frac{m^{6}+7m^{4}+49m^{2}-7m^{4}-49m^{2}-343-m^{2}+343}{m^{4}+7m^{2}+49}$$

Combine like terms in m^{6}+7m^{4}+49m^{2}-7m^{4}-49m^{2}-343-m^{2}+343.

$$\frac{m^{6}-m^{2}}{m^{4}+7m^{2}+49}$$

Expand

$$\frac{m^{6}-m^{2}}{m^{4}+7m^{2}+49}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{m^{4}-49}{m^{2}+7}.

$$\frac{\left(m^{2}-7\right)\left(m^{2}+7\right)}{m^{2}+7}-\frac{m^{2}-343}{m^{4}+7m^{2}+49}$$

Cancel out m^{2}+7 in both numerator and denominator.

$$m^{2}-7-\frac{m^{2}-343}{m^{4}+7m^{2}+49}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply m^{2}-7 times \frac{m^{4}+7m^{2}+49}{m^{4}+7m^{2}+49}.

$$\frac{\left(m^{2}-7\right)\left(m^{4}+7m^{2}+49\right)}{m^{4}+7m^{2}+49}-\frac{m^{2}-343}{m^{4}+7m^{2}+49}$$

Since \frac{\left(m^{2}-7\right)\left(m^{4}+7m^{2}+49\right)}{m^{4}+7m^{2}+49} and \frac{m^{2}-343}{m^{4}+7m^{2}+49} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(m^{2}-7\right)\left(m^{4}+7m^{2}+49\right)-\left(m^{2}-343\right)}{m^{4}+7m^{2}+49}$$

Do the multiplications in \left(m^{2}-7\right)\left(m^{4}+7m^{2}+49\right)-\left(m^{2}-343\right).

$$\frac{m^{6}+7m^{4}+49m^{2}-7m^{4}-49m^{2}-343-m^{2}+343}{m^{4}+7m^{2}+49}$$

Combine like terms in m^{6}+7m^{4}+49m^{2}-7m^{4}-49m^{2}-343-m^{2}+343.

$$\frac{m^{6}-m^{2}}{m^{4}+7m^{2}+49}$$