Solve (1/x^{4-x^{3}}-1/x^4+x^3)*x^2-1/2+1/x^2

$$( \frac { 1 } { x ^ { 4 } - x ^ { 3 } } - \frac { 1 } { x ^ { 4 } + x ^ { 3 } } ) \cdot \frac { x ^ { 2 } - 1 } { 2 } + \frac { 1 } { x ^ { 2 } }$$

Evaluate

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

Short Solution Steps

Factor x^{4}-x^{3}. Factor x^{4}+x^{3}.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-1\right)x^{3} and \left(x+1\right)x^{3} is \left(x-1\right)\left(x+1\right)x^{3}. Multiply \frac{1}{\left(x-1\right)x^{3}} times \frac{x+1}{x+1}. Multiply \frac{1}{\left(x+1\right)x^{3}} times \frac{x-1}{x-1}.

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

Since \frac{x+1}{\left(x-1\right)\left(x+1\right)x^{3}} and \frac{x-1}{\left(x-1\right)\left(x+1\right)x^{3}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in x+1-\left(x-1\right).

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

Combine like terms in x+1-x+1.

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

Multiply \frac{2}{\left(x-1\right)\left(x+1\right)x^{3}} times \frac{x^{2}-1}{2} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out 2 in both numerator and denominator.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-1\right)\left(x+1\right)x^{3} and x^{2} is \left(x-1\right)\left(x+1\right)x^{3}. Multiply \frac{1}{x^{2}} times \frac{x\left(x-1\right)\left(x+1\right)}{x\left(x-1\right)\left(x+1\right)}.

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

Since \frac{x^{2}-1}{\left(x-1\right)\left(x+1\right)x^{3}} and \frac{x\left(x-1\right)\left(x+1\right)}{\left(x-1\right)\left(x+1\right)x^{3}} have the same denominator, add them by adding their numerators.

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

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

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

Combine like terms in x^{2}-1+x^{3}+x^{2}-x^{2}-x.

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

Factor the expressions that are not already factored in \frac{x^{2}-1+x^{3}-x}{\left(x-1\right)\left(x+1\right)x^{3}}.

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

Cancel out \left(x-1\right)\left(x+1\right) in both numerator and denominator.

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

Expand

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

Short Solution Steps

Factor x^{4}-x^{3}. Factor x^{4}+x^{3}.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-1\right)x^{3} and \left(x+1\right)x^{3} is \left(x-1\right)\left(x+1\right)x^{3}. Multiply \frac{1}{\left(x-1\right)x^{3}} times \frac{x+1}{x+1}. Multiply \frac{1}{\left(x+1\right)x^{3}} times \frac{x-1}{x-1}.

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

Since \frac{x+1}{\left(x-1\right)\left(x+1\right)x^{3}} and \frac{x-1}{\left(x-1\right)\left(x+1\right)x^{3}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in x+1-\left(x-1\right).

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

Combine like terms in x+1-x+1.

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

Multiply \frac{2}{\left(x-1\right)\left(x+1\right)x^{3}} times \frac{x^{2}-1}{2} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out 2 in both numerator and denominator.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-1\right)\left(x+1\right)x^{3} and x^{2} is \left(x-1\right)\left(x+1\right)x^{3}. Multiply \frac{1}{x^{2}} times \frac{x\left(x-1\right)\left(x+1\right)}{x\left(x-1\right)\left(x+1\right)}.

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

Since \frac{x^{2}-1}{\left(x-1\right)\left(x+1\right)x^{3}} and \frac{x\left(x-1\right)\left(x+1\right)}{\left(x-1\right)\left(x+1\right)x^{3}} have the same denominator, add them by adding their numerators.

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

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

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

Combine like terms in x^{2}-1+x^{3}+x^{2}-x^{2}-x.

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

Factor the expressions that are not already factored in \frac{x^{2}-1+x^{3}-x}{\left(x-1\right)\left(x+1\right)x^{3}}.

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

Cancel out \left(x-1\right)\left(x+1\right) in both numerator and denominator.

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