Solve 1-2/x^2+1/x^4/1+frac{2{x}+1/x^2}

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

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x^{2}}{x^{2}}.

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

Since \frac{x^{2}}{x^{2}} and \frac{2}{x^{2}} have the same denominator, subtract them by subtracting their numerators.

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

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

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

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

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x}{x}.

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

Since \frac{x}{x} and \frac{2}{x} have the same denominator, add them by adding their numerators.

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

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

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

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

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

Do the multiplications in \left(x+2\right)x+1.

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

Divide \frac{x^{4}-2x^{2}+1}{x^{4}} by \frac{x^{2}+2x+1}{x^{2}} by multiplying \frac{x^{4}-2x^{2}+1}{x^{4}} by the reciprocal of \frac{x^{2}+2x+1}{x^{2}}.

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

Cancel out x^{2} in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Cancel out \left(x+1\right)^{2} in both numerator and denominator.

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

Expand the expression.

$$\frac{x^{2}-2x+1}{x^{2}}$$

Expand

$$1-\frac{2}{x}+\frac{1}{x^{2}}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x^{2}}{x^{2}}.

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

Since \frac{x^{2}}{x^{2}} and \frac{2}{x^{2}} have the same denominator, subtract them by subtracting their numerators.

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

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

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

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

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x}{x}.

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

Since \frac{x}{x} and \frac{2}{x} have the same denominator, add them by adding their numerators.

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

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

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

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

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

Do the multiplications in \left(x+2\right)x+1.

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

Divide \frac{x^{4}-2x^{2}+1}{x^{4}} by \frac{x^{2}+2x+1}{x^{2}} by multiplying \frac{x^{4}-2x^{2}+1}{x^{4}} by the reciprocal of \frac{x^{2}+2x+1}{x^{2}}.

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

Cancel out x^{2} in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Cancel out \left(x+1\right)^{2} in both numerator and denominator.

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

Expand the expression.

$$\frac{x^{2}-2x+1}{x^{2}}$$