Answer :

To solve the problem [tex]\(\frac{14}{15} + \frac{8}{9}\)[/tex], we need to find a common denominator so we can add the fractions together.

1. Find the Least Common Denominator (LCD):

The denominators are 15 and 9. To find the least common denominator, determine the least common multiple (LCM) of these two numbers.

- The prime factorization of 15 is [tex]\(3 \times 5\)[/tex].
- The prime factorization of 9 is [tex]\(3^2\)[/tex].

The LCM is the highest power of each prime that appears in these factorizations. So, we take:
- [tex]\(3^2\)[/tex] (because it is the highest power of 3 appearing)
- [tex]\(5^1\)[/tex] (because it appears in 15)

Therefore, the LCM of 15 and 9 is [tex]\(3^2 \times 5 = 9 \times 5 = 45\)[/tex].

2. Convert each fraction to the common denominator of 45:

- For [tex]\(\frac{14}{15}\)[/tex]:
[tex]\[
\frac{14}{15} = \frac{14 \times 3}{15 \times 3} = \frac{42}{45}
\][/tex]

- For [tex]\(\frac{8}{9}\)[/tex]:
[tex]\[
\frac{8}{9} = \frac{8 \times 5}{9 \times 5} = \frac{40}{45}
\][/tex]

3. Add the fractions:

Now that both fractions have the same denominator, simply add their numerators:

[tex]\[
\frac{42}{45} + \frac{40}{45} = \frac{42 + 40}{45} = \frac{82}{45}
\][/tex]

The result of [tex]\(\frac{14}{15} + \frac{8}{9}\)[/tex] is [tex]\(\frac{82}{45}\)[/tex].

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