Solve frac{y^{2}-2y}{y^2+y-2}*y-1/y^2+4y+4divy^2+y-6/y^6

$$\frac { y ^ { 2 } - 2 y } { y ^ { 2 } + y - 2 } \cdot \frac { y - 1 } { y ^ { 2 } + 4 y + 4 } \div \frac { y ^ { 2 } + y - 6 } { y ^ { 6 } }$$

Evaluate

$$\frac{y^{7}}{\left(y+3\right)\left(y+2\right)^{3}}$$

Short Solution Steps

Multiply \frac{y^{2}-2y}{y^{2}+y-2} times \frac{y-1}{y^{2}+4y+4} by multiplying numerator times numerator and denominator times denominator.

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

Divide \frac{\left(y^{2}-2y\right)\left(y-1\right)}{\left(y^{2}+y-2\right)\left(y^{2}+4y+4\right)} by \frac{y^{2}+y-6}{y^{6}} by multiplying \frac{\left(y^{2}-2y\right)\left(y-1\right)}{\left(y^{2}+y-2\right)\left(y^{2}+4y+4\right)} by the reciprocal of \frac{y^{2}+y-6}{y^{6}}.

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

Factor the expressions that are not already factored.

$$\frac{y\left(y-2\right)\left(y-1\right)y^{6}}{\left(y-2\right)\left(y-1\right)\left(y+2\right)\left(y+3\right)\left(y+2\right)^{2}}$$

Cancel out \left(y-2\right)\left(y-1\right) in both numerator and denominator.

$$\frac{yy^{6}}{\left(y+2\right)\left(y+3\right)\left(y+2\right)^{2}}$$

Expand the expression.

$$\frac{y^{7}}{y^{4}+9y^{3}+30y^{2}+44y+24}$$

Expand

$$\frac{y^{7}}{\left(y+2\right)\left(y+3\right)\left(y^{2}+4y+4\right)}$$

Short Solution Steps

Multiply \frac{y^{2}-2y}{y^{2}+y-2} times \frac{y-1}{y^{2}+4y+4} by multiplying numerator times numerator and denominator times denominator.

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

Divide \frac{\left(y^{2}-2y\right)\left(y-1\right)}{\left(y^{2}+y-2\right)\left(y^{2}+4y+4\right)} by \frac{y^{2}+y-6}{y^{6}} by multiplying \frac{\left(y^{2}-2y\right)\left(y-1\right)}{\left(y^{2}+y-2\right)\left(y^{2}+4y+4\right)} by the reciprocal of \frac{y^{2}+y-6}{y^{6}}.

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

Factor the expressions that are not already factored.

$$\frac{y\left(y-2\right)\left(y-1\right)y^{6}}{\left(y-2\right)\left(y-1\right)\left(y+2\right)\left(y+3\right)\left(y+2\right)^{2}}$$

Cancel out \left(y-2\right)\left(y-1\right) in both numerator and denominator.

$$\frac{yy^{6}}{\left(y+2\right)\left(y+3\right)\left(y+2\right)^{2}}$$

Expand the expression.

$$\frac{y^{7}}{y^{4}+9y^{3}+30y^{2}+44y+24}$$