Simplify. Write your answer using whole numbers and variables. $\newline$ $\frac{r - 1}{r^2 - r}$

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Algebra 1

Evaluate expressions using exponent rules

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Q. Simplify. Write your answer using whole numbers and variables.

\newline

\frac{r - 1}{r^2 - r}

Identify Terms and Operation: Identify the terms and the operation to be performed. We have an expression with a variable $r$ , and we need to simplify it by performing the subtraction and division operations.
Factor Denominator: Factor the denominator of the fraction. $\newline$ The denominator $r^2 - r$ can be factored by taking out the common factor $r$ . $\newline$ $r^2 - r = r(r - 1)$
Rewrite with Factored Denominator: Rewrite the expression with the factored denominator. $\newline$ The expression becomes $r - \frac{1}{r(r - 1)}$ .
Identify Common Denominator: Identify the common denominator to combine the terms. $\newline$ The common denominator for the terms is $r(r - 1)$ .
Rewrite First Term as Fraction: Rewrite the first term $r$ as a fraction over the common denominator. $r$ can be written as $\frac{r}{1}$ , and to have the common denominator, we multiply both the numerator and the denominator by $r(r - 1)$ . $r = \frac{r \times r(r - 1)}{r(r - 1)} = \frac{r^2(r - 1)}{r(r - 1)}$
Combine Terms Over Common Denominator: Combine the terms over the common denominator. $\newline$ Now we have $r^2(r - 1) / (r(r - 1)) - 1/(r(r - 1))$ . $\newline$ We can combine these over the common denominator: $\newline$ $(r^2(r - 1) - 1) / (r(r - 1))$
Distribute $r^2$ in Numerator: Distribute the $r^2$ across the $(r - 1)$ in the numerator. $\newline$ $r^2(r - 1) = r^3 - r^2$ $\newline$ Now the expression is $(r^3 - r^2 - 1) / (r(r - 1))$ .
Simplify Expression: Simplify the expression if possible. $\newline$ The expression $(r^3 - r^2 - 1) / (r(r - 1))$ is already simplified as there are no common factors that can be canceled between the numerator and the denominator.