Solve (x^2+1/x^2)^2-2

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

Evaluate

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

Short Solution Steps

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

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

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

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

Do the multiplications in x^{2}x^{2}+1.

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

To raise \frac{x^{4}+1}{x^{2}} to a power, raise both numerator and denominator to the power and then divide.

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

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

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

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

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

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

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

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

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

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

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

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

$$\frac{x^{8}+2x^{4}+1-2x^{4}}{x^{4}}$$

Combine like terms in x^{8}+2x^{4}+1-2x^{4}.

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

Expand

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

Short Solution Steps

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

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

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

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

Do the multiplications in x^{2}x^{2}+1.

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

To raise \frac{x^{4}+1}{x^{2}} to a power, raise both numerator and denominator to the power and then divide.

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

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

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

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

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

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

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

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

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

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

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

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

$$\frac{x^{8}+2x^{4}+1-2x^{4}}{x^{4}}$$

Combine like terms in x^{8}+2x^{4}+1-2x^{4}.

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