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Simplify. Write your answer using whole numbers and variables.\newliner1r2r\frac{r - 1}{r^2 - r}

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Q. Simplify. Write your answer using whole numbers and variables.\newliner1r2r\frac{r - 1}{r^2 - r}
  1. Identify Terms and Operation: Identify the terms and the operation to be performed. We have an expression with a variable rr, and we need to simplify it by performing the subtraction and division operations.
  2. Factor Denominator: Factor the denominator of the fraction.\newlineThe denominator r2rr^2 - r can be factored by taking out the common factor rr.\newliner2r=r(r1)r^2 - r = r(r - 1)
  3. Rewrite with Factored Denominator: Rewrite the expression with the factored denominator.\newlineThe expression becomes r1r(r1)r - \frac{1}{r(r - 1)}.
  4. Identify Common Denominator: Identify the common denominator to combine the terms.\newlineThe common denominator for the terms is r(r1)r(r - 1).
  5. Rewrite First Term as Fraction: Rewrite the first term rr as a fraction over the common denominator.rr can be written as r1\frac{r}{1}, and to have the common denominator, we multiply both the numerator and the denominator by r(r1)r(r - 1).r=r×r(r1)r(r1)=r2(r1)r(r1)r = \frac{r \times r(r - 1)}{r(r - 1)} = \frac{r^2(r - 1)}{r(r - 1)}
  6. Combine Terms Over Common Denominator: Combine the terms over the common denominator.\newlineNow we have r2(r1)/(r(r1))1/(r(r1))r^2(r - 1) / (r(r - 1)) - 1/(r(r - 1)).\newlineWe can combine these over the common denominator:\newline(r2(r1)1)/(r(r1))(r^2(r - 1) - 1) / (r(r - 1))
  7. Distribute r2r^2 in Numerator: Distribute the r2r^2 across the (r1)(r - 1) in the numerator.\newliner2(r1)=r3r2r^2(r - 1) = r^3 - r^2\newlineNow the expression is (r3r21)/(r(r1))(r^3 - r^2 - 1) / (r(r - 1)).
  8. Simplify Expression: Simplify the expression if possible.\newlineThe expression (r3r21)/(r(r1))(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.