Solve 1-3*(1/y)^2+(1/y)^3/frac{1{y}*(1-1/y)}

$$\frac { 1 - 3 \cdot ( \frac { 1 } { y } ) ^ { 2 } + ( \frac { 1 } { y } ) ^ { 3 } } { \frac { 1 } { y } \cdot ( 1 - \frac { 1 } { y } ) }$$

Evaluate

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

Short Solution Steps

To raise \frac{1}{y} to a power, raise both numerator and denominator to the power and then divide.

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

Express 3\times \frac{1^{2}}{y^{2}} as a single fraction.

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

Calculate 1 to the power of 2 and get 1.

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

Multiply 3 and 1 to get 3.

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

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

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

Since \frac{y^{2}}{y^{2}} and \frac{3}{y^{2}} have the same denominator, subtract them by subtracting their numerators.

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

To raise \frac{1}{y} to a power, raise both numerator and denominator to the power and then divide.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y^{2} and y^{3} is y^{3}. Multiply \frac{y^{2}-3}{y^{2}} times \frac{y}{y}.

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

Since \frac{\left(y^{2}-3\right)y}{y^{3}} and \frac{1^{3}}{y^{3}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(y^{2}-3\right)y+1^{3}.

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

Combine like terms in y^{3}-3y+1^{3}.

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

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

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

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

$$\frac{\frac{y^{3}-3y+1}{y^{3}}}{\frac{1}{y}\times \frac{y-1}{y}}$$

Multiply \frac{1}{y} times \frac{y-1}{y} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{y^{3}-3y+1}{y^{3}}}{\frac{y-1}{yy}}$$

Divide \frac{y^{3}-3y+1}{y^{3}} by \frac{y-1}{yy} by multiplying \frac{y^{3}-3y+1}{y^{3}} by the reciprocal of \frac{y-1}{yy}.

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

Cancel out yy in both numerator and denominator.

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

Use the distributive property to multiply y by y-1.

$$\frac{y^{3}-3y+1}{y^{2}-y}$$

Expand

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

Short Solution Steps

To raise \frac{1}{y} to a power, raise both numerator and denominator to the power and then divide.

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

Express 3\times \frac{1^{2}}{y^{2}} as a single fraction.

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

Calculate 1 to the power of 2 and get 1.

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

Multiply 3 and 1 to get 3.

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

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

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

Since \frac{y^{2}}{y^{2}} and \frac{3}{y^{2}} have the same denominator, subtract them by subtracting their numerators.

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

To raise \frac{1}{y} to a power, raise both numerator and denominator to the power and then divide.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y^{2} and y^{3} is y^{3}. Multiply \frac{y^{2}-3}{y^{2}} times \frac{y}{y}.

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

Since \frac{\left(y^{2}-3\right)y}{y^{3}} and \frac{1^{3}}{y^{3}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(y^{2}-3\right)y+1^{3}.

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

Combine like terms in y^{3}-3y+1^{3}.

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

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

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

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

$$\frac{\frac{y^{3}-3y+1}{y^{3}}}{\frac{1}{y}\times \frac{y-1}{y}}$$

Multiply \frac{1}{y} times \frac{y-1}{y} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{y^{3}-3y+1}{y^{3}}}{\frac{y-1}{yy}}$$

Divide \frac{y^{3}-3y+1}{y^{3}} by \frac{y-1}{yy} by multiplying \frac{y^{3}-3y+1}{y^{3}} by the reciprocal of \frac{y-1}{yy}.

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

Cancel out yy in both numerator and denominator.

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

Use the distributive property to multiply y by y-1.

$$\frac{y^{3}-3y+1}{y^{2}-y}$$