91Ó°ÊÓ

The distributive property is a fundamental principle in algebra that allows us to simplify expressions easily. It states that when you multiply a term by a sum or difference inside parentheses, you should multiply that term by each individual term within the parentheses. For example, if we have \(a(b + c)\), we distribute \(a\) to get \(ab + ac\).
In the original exercise, we had \(\text{\sqrt{3}(\text{\text{\sqrt{2x}}\text{ + 5)}})}\). Applying the distributive property means multiplying \(\text{\sqrt{3}}\) by both \(\text{\sqrt{2x}}\) and \(\text{5}}\). This results in the expression \(\text{\text{\sqrt{3}} \text{\cdot} \text{\text{\sqrt{2x}}} + \text{\text{\sqrt{3}} \text{\cdot} \text{5}}}}\).

Square root properties help us simplify expressions involving roots. One important property is that the product of square roots is the square root of the product of their radicands. In other words, \(\text{\text{ \sqrt{a} \text{\cdot} \text{\text{\sqrt{b}}} = \text{\text{\text{ \sqrt{ab}}}}\).
Applying this to our previous step, we should simplify \(\text{\text{\sqrt{3} \text{\cdot} \text{\text{\text{\sqrt{2x}}}}}}\). Using the property \(\text{\sqrt{3}} \text{\text{ \text{\text{\cdot}} \text{2x}}} = \text{\sqrt{6x}}\).
Consequently, the expression transforms into: \(\text{\sqrt{6x}} + 5\text{\text{ \text{\sqrt{3}}}}\).

Combining like terms is a process used to simplify expressions by merging terms that have the same variables raised to the same power. However, in this particular problem, we don't have any like terms to combine. The terms \(\text{\text{\sqrt{6x}}}\) and \(\text{\text{5\sqrt{3}}}\) are not like terms because they involve different radicands and coefficients.
The final simplified expression thus remains: \(\text{\text{\sqrt{6x}} + 5\text{\text{\text{\sqrt{3}}}}\).