To find the least common denominator (LCD) of the fractions \(\frac{13}{x-3}\) and \(\frac{x+10}{3-x}\), we need to analyze the denominators and find a common expression that allows us to combine these fractions.
1. Let's start by examining the denominators of the given fractions:
- The first fraction has the denominator \(x-3\).
- The second fraction has the denominator \(3-x\).
2. Notice that the denominator \(3-x\) in the second fraction can be rewritten:
- We can factor out a \(-1\) from \(3-x\):
[tex]\[
3-x = -1 \cdot (x-3)
\][/tex]
3. Now we see that \(3-x\) is equivalent to \(-1 \cdot (x-3)\):
- This means that \(3-x\) is simply the negative of \(x-3\).
4. Since both denominators are effectively the same up to a sign, we can conclude that the least common denominator must be based on the expression \(x-3\):
- In its simplest form, the least common denominator (LCD) is \(x-3\).
Thus, the least common denominator (LCD) of the fractions \(\frac{13}{x-3}\) and \(\frac{x+10}{3-x}\) is:
[tex]\[
\boxed{x-3}
\][/tex]
