Solve frac{x-16x^{-1}}{5x}*(2-2x^-1-2x^-2/x^-1+4x^-2)

$$\frac { x - 16 x ^ { - 1 } } { 5 x } \cdot ( 2 - \frac { 2 x ^ { - 1 } - 2 x ^ { - 2 } } { x ^ { - 1 } + 4 x ^ { - 2 } } )$$

Evaluate

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

Short Solution Steps

Factor the expressions that are not already factored in \frac{x-16x^{-1}}{5x}.

$$\frac{\frac{1}{x}\left(x-4\right)\left(x+4\right)}{5x}\left(2-\frac{2x^{-1}-2x^{-2}}{x^{-1}+4x^{-2}}\right)$$

To divide powers of the same base, subtract the numerator's exponent from the denominator's exponent.

$$\frac{\left(x-4\right)\left(x+4\right)}{5x^{2}}\left(2-\frac{2x^{-1}-2x^{-2}}{x^{-1}+4x^{-2}}\right)$$

Factor the expressions that are not already factored in \frac{2x^{-1}-2x^{-2}}{x^{-1}+4x^{-2}}.

$$\frac{\left(x-4\right)\left(x+4\right)}{5x^{2}}\left(2-\frac{2\times \left(\frac{1}{x}\right)^{2}\left(x-1\right)}{\left(\frac{1}{x}\right)^{2}\left(x+4\right)}\right)$$

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

$$\frac{\left(x-4\right)\left(x+4\right)}{5x^{2}}\left(2-\frac{2\left(x-1\right)}{x+4}\right)$$

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

$$\frac{\left(x-4\right)\left(x+4\right)}{5x^{2}}\left(\frac{2\left(x+4\right)}{x+4}-\frac{2\left(x-1\right)}{x+4}\right)$$

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

$$\frac{\left(x-4\right)\left(x+4\right)}{5x^{2}}\times \frac{2\left(x+4\right)-2\left(x-1\right)}{x+4}$$

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

$$\frac{\left(x-4\right)\left(x+4\right)}{5x^{2}}\times \frac{2x+8-2x+2}{x+4}$$

Combine like terms in 2x+8-2x+2.

$$\frac{\left(x-4\right)\left(x+4\right)}{5x^{2}}\times \frac{10}{x+4}$$

Multiply \frac{\left(x-4\right)\left(x+4\right)}{5x^{2}} times \frac{10}{x+4} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(x-4\right)\left(x+4\right)\times 10}{5x^{2}\left(x+4\right)}$$

Cancel out 5\left(x+4\right) in both numerator and denominator.

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

Use the distributive property to multiply 2 by x-4.

$$\frac{2x-8}{x^{2}}$$

Expand

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

Short Solution Steps

Factor the expressions that are not already factored in \frac{x-16x^{-1}}{5x}.

$$\frac{\frac{1}{x}\left(x-4\right)\left(x+4\right)}{5x}\left(2-\frac{2x^{-1}-2x^{-2}}{x^{-1}+4x^{-2}}\right)$$

To divide powers of the same base, subtract the numerator's exponent from the denominator's exponent.

$$\frac{\left(x-4\right)\left(x+4\right)}{5x^{2}}\left(2-\frac{2x^{-1}-2x^{-2}}{x^{-1}+4x^{-2}}\right)$$

Factor the expressions that are not already factored in \frac{2x^{-1}-2x^{-2}}{x^{-1}+4x^{-2}}.

$$\frac{\left(x-4\right)\left(x+4\right)}{5x^{2}}\left(2-\frac{2\times \left(\frac{1}{x}\right)^{2}\left(x-1\right)}{\left(\frac{1}{x}\right)^{2}\left(x+4\right)}\right)$$

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

$$\frac{\left(x-4\right)\left(x+4\right)}{5x^{2}}\left(2-\frac{2\left(x-1\right)}{x+4}\right)$$

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

$$\frac{\left(x-4\right)\left(x+4\right)}{5x^{2}}\left(\frac{2\left(x+4\right)}{x+4}-\frac{2\left(x-1\right)}{x+4}\right)$$

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

$$\frac{\left(x-4\right)\left(x+4\right)}{5x^{2}}\times \frac{2\left(x+4\right)-2\left(x-1\right)}{x+4}$$

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

$$\frac{\left(x-4\right)\left(x+4\right)}{5x^{2}}\times \frac{2x+8-2x+2}{x+4}$$

Combine like terms in 2x+8-2x+2.

$$\frac{\left(x-4\right)\left(x+4\right)}{5x^{2}}\times \frac{10}{x+4}$$

Multiply \frac{\left(x-4\right)\left(x+4\right)}{5x^{2}} times \frac{10}{x+4} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(x-4\right)\left(x+4\right)\times 10}{5x^{2}\left(x+4\right)}$$

Cancel out 5\left(x+4\right) in both numerator and denominator.

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

Use the distributive property to multiply 2 by x-4.

$$\frac{2x-8}{x^{2}}$$