To find the common denominator for [tex]\(\frac{3}{4}\)[/tex] and [tex]\(\frac{7}{9}\)[/tex], we need to determine the Least Common Multiple (LCM) of the denominators 4 and 9.
First, let's factorize the denominators:
- The prime factorization of 4 is [tex]\(2^2\)[/tex].
- The prime factorization of 9 is [tex]\(3^2\)[/tex].
Next, to find the LCM, we take the highest power of each prime factor that appears in these factorizations:
- For the prime factor 2, the highest power appearing is [tex]\(2^2\)[/tex].
- For the prime factor 3, the highest power appearing is [tex]\(3^2\)[/tex].
We then multiply these together:
[tex]\[
\text{LCM} = 2^2 \times 3^2 = 4 \times 9 = 36
\][/tex]
Therefore, the least common denominator for [tex]\(\frac{3}{4}\)[/tex] and [tex]\(\frac{7}{9}\)[/tex] is 36.
The answer is 36.
