Certainly! Let's solve the expression [tex]\(\frac{3a}{bc} + \frac{2b}{ac}\)[/tex] step-by-step.
1. Identify the common denominator:
- The denominators of each fraction are [tex]\(bc\)[/tex] and [tex]\(ac\)[/tex], respectively.
- To add these fractions, we need a common denominator.
- The least common multiple (LCM) of [tex]\(bc\)[/tex] and [tex]\(ac\)[/tex] is [tex]\(abc\)[/tex].
2. Rewrite each fraction with the common denominator:
- For the first term [tex]\(\frac{3a}{bc}\)[/tex]:
- Multiply the numerator and the denominator by [tex]\(a\)[/tex]:
[tex]\[
\frac{3a \cdot a}{bc \cdot a} = \frac{3a^2}{abc}
\][/tex]
- For the second term [tex]\(\frac{2b}{ac}\)[/tex]:
- Multiply the numerator and the denominator by [tex]\(b\)[/tex]:
[tex]\[
\frac{2b \cdot b}{ac \cdot b} = \frac{2b^2}{abc}
\][/tex]
3. Sum the rewritten fractions:
- Now that both fractions have the same denominator, add the numerators together:
[tex]\[
\frac{3a^2}{abc} + \frac{2b^2}{abc} = \frac{3a^2 + 2b^2}{abc}
\][/tex]
4. Simplify if possible (although in this case, the expression is already simplified).
So the final simplified expression is:
[tex]\[
\frac{3a^2 + 2b^2}{abc}
\][/tex]
Comparing this with the options given:
- A. [tex]\(\frac{3a^2 + 2b^2}{abc^2}\)[/tex] — Incorrect
- B. [tex]\(\frac{3a^2 + 2b^2}{abc}\)[/tex] — Correct
- C. [tex]\(\frac{3a^2c + 2b^3c}{abc^2}\)[/tex] — Incorrect
- D. [tex]\(\frac{3b^2 + 2a^2}{abc}\)[/tex] — Incorrect
The correct answer is B. [tex]\(\frac{3a^2 + 2b^2}{abc}\)[/tex].
