12. A. $$ \frac{2+2 c^{4}}{1-c^{4}}+\frac{4 c^{2}}{1+c^{4}}+\frac{8 c^{6}}{1-c^{8}} $$

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UpStudy AI · Accepted Answer

Calculate or simplify the expression $$ (2+2*c^4)/(1-c^4)+(4*c^2)/(1+c^4)+(8*c^6)/(1-c^8) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{\left(2+2c^{4}\right)}{\left(1-c^{4}\right)}+\frac{4c^{2}}{\left(1+c^{4}\right)}+\frac{8c^{6}}{\left(1-c^{8}\right)}$$
- step1: Remove the parentheses:
  $$\frac{2+2c^{4}}{1-c^{4}}+\frac{4c^{2}}{1+c^{4}}+\frac{8c^{6}}{1-c^{8}}$$
- step2: Rewrite the fractions:
  $$\frac{2+2c^{4}}{1-c^{4}}+\frac{4c^{2}}{1+c^{4}}-\frac{8c^{6}}{-1+c^{8}}$$
- step3: Factor the expression:
  $$\frac{2+2c^{4}}{1-c^{4}}+\frac{4c^{2}}{1+c^{4}}-\frac{8c^{6}}{\left(c^{4}+1\right)\left(c^{4}-1\right)}$$
- step4: Reduce fractions to a common denominator:
  $$\frac{\left(2+2c^{4}\right)\left(1+c^{4}\right)}{\left(1-c^{4}\right)\left(1+c^{4}\right)}+\frac{4c^{2}\left(1-c^{4}\right)}{\left(1+c^{4}\right)\left(1-c^{4}\right)}-\frac{8c^{6}\left(-1\right)}{\left(c^{4}+1\right)\left(c^{4}-1\right)\left(-1\right)}$$
- step5: Reorder the terms:
  $$\frac{\left(2+2c^{4}\right)\left(1+c^{4}\right)}{\left(1-c^{4}\right)\left(1+c^{4}\right)}+\frac{4c^{2}\left(1-c^{4}\right)}{\left(1+c^{4}\right)\left(1-c^{4}\right)}-\frac{8c^{6}\left(-1\right)}{-\left(c^{4}+1\right)\left(c^{4}-1\right)}$$
- step6: Rewrite the expression:
  $$\frac{\left(2+2c^{4}\right)\left(1+c^{4}\right)}{\left(1-c^{4}\right)\left(1+c^{4}\right)}+\frac{4c^{2}\left(1-c^{4}\right)}{\left(1-c^{4}\right)\left(1+c^{4}\right)}-\frac{8c^{6}\left(-1\right)}{\left(1-c^{4}\right)\left(1+c^{4}\right)}$$
- step7: Transform the expression:
  $$\frac{\left(2+2c^{4}\right)\left(1+c^{4}\right)+4c^{2}\left(1-c^{4}\right)-8c^{6}\left(-1\right)}{\left(1-c^{4}\right)\left(1+c^{4}\right)}$$
- step8: Multiply the terms:
  $$\frac{2+4c^{4}+2c^{8}+4c^{2}\left(1-c^{4}\right)-8c^{6}\left(-1\right)}{\left(1-c^{4}\right)\left(1+c^{4}\right)}$$
- step9: Multiply the terms:
  $$\frac{2+4c^{4}+2c^{8}+4c^{2}-4c^{6}-8c^{6}\left(-1\right)}{\left(1-c^{4}\right)\left(1+c^{4}\right)}$$
- step10: Rewrite the expression:
  $$\frac{2+4c^{4}+2c^{8}+4c^{2}-4c^{6}-\left(-8c^{6}\right)}{\left(1-c^{4}\right)\left(1+c^{4}\right)}$$
- step11: Subtract the terms:
  $$\frac{2+4c^{4}+2c^{8}+4c^{2}+4c^{6}}{\left(1-c^{4}\right)\left(1+c^{4}\right)}$$
- step12: Factor the expression:
  $$\frac{\left(c^{2}+1\right)\left(2c^{2}+2c^{6}+2+2c^{4}\right)}{\left(c^{2}+1\right)\left(1-c^{2}\right)\left(1+c^{4}\right)}$$
- step13: Reduce the fraction:
  $$\frac{2c^{2}+2c^{6}+2+2c^{4}}{\left(1-c^{2}\right)\left(1+c^{4}\right)}$$
- step14: Factor the expression:
  $$\frac{\left(c^{4}+1\right)\left(2c^{2}+2\right)}{\left(1-c^{2}\right)\left(c^{4}+1\right)}$$
- step15: Reduce the fraction:
  $$\frac{2c^{2}+2}{1-c^{2}}$$
The simplified expression is $$ \frac{2c^{2}+2}{1-c^{2}} $$.
The simplified expression is $$ \frac{2c^{2}+2}{1-c^{2}} $$.

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