To simplify the expression
[tex]\[
\frac{\sqrt{75a} + \sqrt{12a} - \sqrt{27a}}{\sqrt{3a}},
\][/tex]
let's break it down step by step.
1. Simplify the individual square roots in the numerator:
- [tex]\(\sqrt{75a}\)[/tex]
[tex]\[
\sqrt{75a} = \sqrt{25 \cdot 3a} = \sqrt{25} \cdot \sqrt{3a} = 5 \sqrt{3a}
\][/tex]
- [tex]\(\sqrt{12a}\)[/tex]
[tex]\[
\sqrt{12a} = \sqrt{4 \cdot 3a} = \sqrt{4} \cdot \sqrt{3a} = 2 \sqrt{3a}
\][/tex]
- [tex]\(\sqrt{27a}\)[/tex]
[tex]\[
\sqrt{27a} = \sqrt{9 \cdot 3a} = \sqrt{9} \cdot \sqrt{3a} = 3 \sqrt{3a}
\][/tex]
2. Substitute these simplified forms back into the numerator:
[tex]\[
5 \sqrt{3a} + 2 \sqrt{3a} - 3 \sqrt{3a}
\][/tex]
3. Combine like terms:
[tex]\[
5 \sqrt{3a} + 2 \sqrt{3a} - 3 \sqrt{3a} = (5 + 2 - 3) \sqrt{3a} = 4 \sqrt{3a}
\][/tex]
4. The original expression is now:
[tex]\[
\frac{4 \sqrt{3a}}{\sqrt{3a}}
\][/tex]
5. Simplify the fraction:
[tex]\[
\frac{4 \sqrt{3a}}{\sqrt{3a}} = 4
\][/tex]
Therefore, the simplified form of the given expression is:
[tex]\[
\boxed{4}
\][/tex]
