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

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

Evaluate

$$\frac{\left(x^{4}-1\right)\left(\left(x^{4}+1\right)^{2}-x^{4}\right)}{x^{6}}$$

Short Solution Steps

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

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

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

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

Do the multiplications in xx+1.

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

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

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

Since \frac{xx}{x} and \frac{1}{x} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in xx-1.

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

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

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

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

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

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

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

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

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

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

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

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

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

Multiply \frac{x^{2}+1}{x} times \frac{x^{2}-1}{x} by multiplying numerator times numerator and denominator times denominator.

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

Multiply \frac{\left(x^{2}+1\right)\left(x^{2}-1\right)}{xx} times \frac{x^{4}-x^{2}+1}{x^{2}} by multiplying numerator times numerator and denominator times denominator.

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

Multiply \frac{\left(x^{2}+1\right)\left(x^{2}-1\right)\left(x^{4}-x^{2}+1\right)}{xxx^{2}} times \frac{x^{4}+x^{2}+1}{x^{2}} by multiplying numerator times numerator and denominator times denominator.

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

Multiply x and x to get x^{2}.

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

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

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

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

$$\frac{\left(x^{2}+1\right)\left(x^{2}-1\right)\left(x^{4}-x^{2}+1\right)\left(x^{4}+x^{2}+1\right)}{x^{6}}$$

Use the distributive property to multiply x^{2}+1 by x^{2}-1 and combine like terms.

$$\frac{\left(x^{4}-1\right)\left(x^{4}-x^{2}+1\right)\left(x^{4}+x^{2}+1\right)}{x^{6}}$$

Use the distributive property to multiply x^{4}-1 by x^{4}-x^{2}+1 and combine like terms.

$$\frac{\left(x^{8}-x^{6}+x^{2}-1\right)\left(x^{4}+x^{2}+1\right)}{x^{6}}$$

Use the distributive property to multiply x^{8}-x^{6}+x^{2}-1 by x^{4}+x^{2}+1 and combine like terms.

$$\frac{x^{12}-1}{x^{6}}$$

Expand

$$x^{6}-\frac{1}{x^{6}}$$

Short Solution Steps

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

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

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

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

Do the multiplications in xx+1.

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

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

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

Since \frac{xx}{x} and \frac{1}{x} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in xx-1.

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

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

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

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

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

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

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

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

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

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

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

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

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

Multiply \frac{x^{2}+1}{x} times \frac{x^{2}-1}{x} by multiplying numerator times numerator and denominator times denominator.

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

Multiply \frac{\left(x^{2}+1\right)\left(x^{2}-1\right)}{xx} times \frac{x^{4}-x^{2}+1}{x^{2}} by multiplying numerator times numerator and denominator times denominator.

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

Multiply \frac{\left(x^{2}+1\right)\left(x^{2}-1\right)\left(x^{4}-x^{2}+1\right)}{xxx^{2}} times \frac{x^{4}+x^{2}+1}{x^{2}} by multiplying numerator times numerator and denominator times denominator.

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

Multiply x and x to get x^{2}.

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

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

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

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

$$\frac{\left(x^{2}+1\right)\left(x^{2}-1\right)\left(x^{4}-x^{2}+1\right)\left(x^{4}+x^{2}+1\right)}{x^{6}}$$

Use the distributive property to multiply x^{2}+1 by x^{2}-1 and combine like terms.

$$\frac{\left(x^{4}-1\right)\left(x^{4}-x^{2}+1\right)\left(x^{4}+x^{2}+1\right)}{x^{6}}$$

Use the distributive property to multiply x^{4}-1 by x^{4}-x^{2}+1 and combine like terms.

$$\frac{\left(x^{8}-x^{6}+x^{2}-1\right)\left(x^{4}+x^{2}+1\right)}{x^{6}}$$

Use the distributive property to multiply x^{8}-x^{6}+x^{2}-1 by x^{4}+x^{2}+1 and combine like terms.

$$\frac{x^{12}-1}{x^{6}}$$