Solve 8/x^9*x^-6+1+8/x^6*x^-2+1

$$\frac { 8 } { x ^ { 9 } \cdot x ^ { - 6 } + 1 } + \frac { 8 } { x ^ { 6 } \cdot x ^ { - 2 } + 1 }$$

Evaluate

$$\frac{8\left(x^{4}+x^{3}+2\right)}{\left(x^{3}+1\right)\left(x^{4}+1\right)}$$

Short Solution Steps

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

$$\frac{8}{x^{3}+1}+\frac{8}{x^{6}x^{-2}+1}$$

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

$$\frac{8}{x^{3}+1}+\frac{8}{x^{4}+1}$$

Factor x^{3}+1.

$$\frac{8}{\left(x+1\right)\left(x^{2}-x+1\right)}+\frac{8}{x^{4}+1}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+1\right)\left(x^{2}-x+1\right) and x^{4}+1 is \left(x+1\right)\left(x^{2}-x+1\right)\left(x^{4}+1\right). Multiply \frac{8}{\left(x+1\right)\left(x^{2}-x+1\right)} times \frac{x^{4}+1}{x^{4}+1}. Multiply \frac{8}{x^{4}+1} times \frac{\left(x+1\right)\left(x^{2}-x+1\right)}{\left(x+1\right)\left(x^{2}-x+1\right)}.

$$\frac{8\left(x^{4}+1\right)}{\left(x+1\right)\left(x^{2}-x+1\right)\left(x^{4}+1\right)}+\frac{8\left(x+1\right)\left(x^{2}-x+1\right)}{\left(x+1\right)\left(x^{2}-x+1\right)\left(x^{4}+1\right)}$$

Since \frac{8\left(x^{4}+1\right)}{\left(x+1\right)\left(x^{2}-x+1\right)\left(x^{4}+1\right)} and \frac{8\left(x+1\right)\left(x^{2}-x+1\right)}{\left(x+1\right)\left(x^{2}-x+1\right)\left(x^{4}+1\right)} have the same denominator, add them by adding their numerators.

$$\frac{8\left(x^{4}+1\right)+8\left(x+1\right)\left(x^{2}-x+1\right)}{\left(x+1\right)\left(x^{2}-x+1\right)\left(x^{4}+1\right)}$$

Do the multiplications in 8\left(x^{4}+1\right)+8\left(x+1\right)\left(x^{2}-x+1\right).

$$\frac{8x^{4}+8+8x^{3}-8x^{2}+8x+8x^{2}-8x+8}{\left(x+1\right)\left(x^{2}-x+1\right)\left(x^{4}+1\right)}$$

Combine like terms in 8x^{4}+8+8x^{3}-8x^{2}+8x+8x^{2}-8x+8.

$$\frac{8x^{4}+16+8x^{3}}{\left(x+1\right)\left(x^{2}-x+1\right)\left(x^{4}+1\right)}$$

Expand \left(x+1\right)\left(x^{2}-x+1\right)\left(x^{4}+1\right).

$$\frac{8x^{4}+16+8x^{3}}{x^{7}+x^{4}+x^{3}+1}$$

Factor

$$\frac{8\left(x^{4}+x^{3}+2\right)}{\left(x+1\right)\left(x^{2}-x+1\right)\left(x^{4}+1\right)}$$