Question

evaluate \frac{cd - 6c + 2}{-5c - 9d^3 + 9d + 10} when c = 2 and d = -2. enter an integer or a fraction in simplest form.

Explanation:

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<reasoning>

Identify the expression and given values

We are given the rational expression:
\[
\frac{cd - 6c + 2}{-5c - 9d^3 + 9d + 10}
\]
We need to evaluate this expression when \(c = 2\) and \(d = -2\).

Evaluate the numerator

Using the Evaluating Rational Expressions knowledge point, we substitute \(c = 2\) and \(d = -2\) into the numerator:
\[

$$\begin{aligned} \text{Numerator} &= cd - 6c + 2 \\ &= (2)(-2) - 6(2) + 2 \\ &= -4 - 12 + 2 \\ &= -14 \end{aligned}$$

\]

Evaluate the denominator

Next, we substitute \(c = 2\) and \(d = -2\) into the denominator:
\[

$$\begin{aligned} \text{Denominator} &= -5c - 9d^3 + 9d + 10 \\ &= -5(2) - 9(-2)^3 + 9(-2) + 10 \\ &= -10 - 9(-8) - 18 + 10 \\ &= -10 + 72 - 18 + 10 \\ &= 54 \end{aligned}$$

\]

Simplify the resulting fraction

Using the Fraction Arithmetic knowledge point, we write the evaluated expression as a fraction and simplify it to its simplest form:
\[
\frac{\text{Numerator}}{\text{Denominator}} = \frac{-14}{54} = \frac{-7 \times 2}{27 \times 2} = -\frac{7}{27}
\]
</reasoning>

<answer>
\(-\frac{7}{27}\)
</answer>

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Answer:

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<reasoning>

Identify the expression and given values

We are given the rational expression:
\[
\frac{cd - 6c + 2}{-5c - 9d^3 + 9d + 10}
\]
We need to evaluate this expression when \(c = 2\) and \(d = -2\).

Evaluate the numerator

Using the Evaluating Rational Expressions knowledge point, we substitute \(c = 2\) and \(d = -2\) into the numerator:
\[

$$\begin{aligned} \text{Numerator} &= cd - 6c + 2 \\ &= (2)(-2) - 6(2) + 2 \\ &= -4 - 12 + 2 \\ &= -14 \end{aligned}$$

\]

Evaluate the denominator

Next, we substitute \(c = 2\) and \(d = -2\) into the denominator:
\[

$$\begin{aligned} \text{Denominator} &= -5c - 9d^3 + 9d + 10 \\ &= -5(2) - 9(-2)^3 + 9(-2) + 10 \\ &= -10 - 9(-8) - 18 + 10 \\ &= -10 + 72 - 18 + 10 \\ &= 54 \end{aligned}$$

\]

Simplify the resulting fraction

Using the Fraction Arithmetic knowledge point, we write the evaluated expression as a fraction and simplify it to its simplest form:
\[
\frac{\text{Numerator}}{\text{Denominator}} = \frac{-14}{54} = \frac{-7 \times 2}{27 \times 2} = -\frac{7}{27}
\]
</reasoning>

<answer>
\(-\frac{7}{27}\)
</answer>

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