Simplify. Express your answer as a single fraction in simplest form \( 9 r-\frac{9 r-6}{r-8} \)

To simplify the expression \( 9 r - \frac{9 r - 6}{r - 8} \), first, we need a common denominator. The common denominator here is \( r - 8 \). Rewriting \( 9r \) with the common denominator, we have: \[ 9 r = \frac{9r(r - 8)}{r - 8} \] Now, substituting this back into the expression gives: \[ \frac{9r(r - 8)}{r - 8} - \frac{9r - 6}{r - 8} \] Combining the fractions over the common denominator: \[ \frac{9r(r - 8) - (9r - 6)}{r - 8} \] Now, simplify the numerator: \[ 9r^2 - 72r - 9r + 6 = 9r^2 - 63r + 6 \] Now, our expression is: \[ \frac{9r^2 - 63r + 6}{r - 8} \] Next, we can factor the numerator. The expression \( 9r^2 - 63r + 6 \) can be factored by taking out the greatest common factor: \[ 3(3r^2 - 21r + 2) \] Next, we can apply the quadratic formula to find the roots of \( 3r^2 - 21r + 2 \): \[ r = \frac{-(-21) \pm \sqrt{(-21)^2 - 4 \cdot 3 \cdot 2}}{2 \cdot 3} = \frac{21 \pm \sqrt{441 - 24}}{6} = \frac{21 \pm \sqrt{417}}{6} \] However, this doesn't appear to factor neatly, so we keep it as it is. Therefore, the simplified form remains: \[ \frac{9r^2 - 63r + 6}{r - 8} \] We need to check if there's any common factors with the denominator \( r - 8 \), but there are none in this case. Therefore, the final answer in simplest form is: \[ \frac{9r^2 - 63r + 6}{r - 8} \] **Final Answer:** \( \frac{9r^2 - 63r + 6}{r - 8} \)