Solve for r 1/r=1/r+1/(r^2)

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

Since ${\displaystyle r,r,r^2}$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part ${\displaystyle 1,1,1}$ then find LCM for the variable part ${\displaystyle r^1,r^1,r^2}$.

The LCM is the smallest positive number that all of the numbers divide into evenly.

The number ${\displaystyle 1}$ is not a prime number because it only has one positive factor, which is itself.

The LCM of ${\displaystyle 1,1,1}$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

The factor for ${\displaystyle r^1}$ is ${\displaystyle r}$ itself.

The factors for ${\displaystyle r^2}$ are ${\displaystyle r\cdot r}$, which is ${\displaystyle r}$ multiplied by each other ${\displaystyle 2}$ times.

The LCM of ${\displaystyle r^1,r^1,r^2}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

Multiply ${\displaystyle r}$ by ${\displaystyle r}$.

Multiply each term in ${\displaystyle \frac{1}{r}=\frac{1}{r}+\frac{1}{r^2}}$ by ${\displaystyle r^2}$.

Simplify the left side.

Simplify the right side.

Move all terms containing ${\displaystyle r}$ to the left side of the equation.

Since ${\displaystyle 0\ne 1}$, there are no solutions.