Solve y^3+1000/y^4-10000/(y^2+100)^-1-10y/y-10

$$\frac { \frac { y ^ { 3 } + 1000 } { y ^ { 4 } - 10000 } } { ( y ^ { 2 } + 100 ) ^ { - 1 } } - \frac { 10 y } { y - 10 }$$

Evaluate

$$y-10$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{y^{3}+1000}{y^{4}-10000}.

$$\frac{\frac{\left(y+10\right)\left(y^{2}-10y+100\right)}{\left(y-10\right)\left(y+10\right)\left(y^{2}+100\right)}}{\left(y^{2}+100\right)^{-1}}-\frac{10y}{y-10}$$

Cancel out y+10 in both numerator and denominator.

$$\frac{\frac{y^{2}-10y+100}{\left(y-10\right)\left(y^{2}+100\right)}}{\left(y^{2}+100\right)^{-1}}-\frac{10y}{y-10}$$

Express \frac{\frac{y^{2}-10y+100}{\left(y-10\right)\left(y^{2}+100\right)}}{\left(y^{2}+100\right)^{-1}} as a single fraction.

$$\frac{y^{2}-10y+100}{\left(y-10\right)\left(y^{2}+100\right)\left(y^{2}+100\right)^{-1}}-\frac{10y}{y-10}$$

Multiply y^{2}+100 and \left(y^{2}+100\right)^{-1} to get 1.

$$\frac{y^{2}-10y+100}{y-10}-\frac{10y}{y-10}$$

Since \frac{y^{2}-10y+100}{y-10} and \frac{10y}{y-10} have the same denominator, subtract them by subtracting their numerators.

$$\frac{y^{2}-10y+100-10y}{y-10}$$

Combine like terms in y^{2}-10y+100-10y.

$$\frac{y^{2}-20y+100}{y-10}$$

Factor the expressions that are not already factored in \frac{y^{2}-20y+100}{y-10}.

$$\frac{\left(y-10\right)^{2}}{y-10}$$

Cancel out y-10 in both numerator and denominator.

$$y-10$$

Expand

$$y-10$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{y^{3}+1000}{y^{4}-10000}.

$$\frac{\frac{\left(y+10\right)\left(y^{2}-10y+100\right)}{\left(y-10\right)\left(y+10\right)\left(y^{2}+100\right)}}{\left(y^{2}+100\right)^{-1}}-\frac{10y}{y-10}$$

Cancel out y+10 in both numerator and denominator.

$$\frac{\frac{y^{2}-10y+100}{\left(y-10\right)\left(y^{2}+100\right)}}{\left(y^{2}+100\right)^{-1}}-\frac{10y}{y-10}$$

Express \frac{\frac{y^{2}-10y+100}{\left(y-10\right)\left(y^{2}+100\right)}}{\left(y^{2}+100\right)^{-1}} as a single fraction.

$$\frac{y^{2}-10y+100}{\left(y-10\right)\left(y^{2}+100\right)\left(y^{2}+100\right)^{-1}}-\frac{10y}{y-10}$$

Multiply y^{2}+100 and \left(y^{2}+100\right)^{-1} to get 1.

$$\frac{y^{2}-10y+100}{y-10}-\frac{10y}{y-10}$$

Since \frac{y^{2}-10y+100}{y-10} and \frac{10y}{y-10} have the same denominator, subtract them by subtracting their numerators.

$$\frac{y^{2}-10y+100-10y}{y-10}$$

Combine like terms in y^{2}-10y+100-10y.

$$\frac{y^{2}-20y+100}{y-10}$$

Factor the expressions that are not already factored in \frac{y^{2}-20y+100}{y-10}.

$$\frac{\left(y-10\right)^{2}}{y-10}$$

Cancel out y-10 in both numerator and denominator.

$$y-10$$