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The distributive property is a fundamental concept in algebra that allows you to multiply a single term by multiple terms inside parentheses. This property states that for any real numbers \(a\), \(b\), and \(c\), the expression \(a(b + c)\) is equal to \(ab + ac\).
In the exercise, we apply the distributive property to multiply each term in the first pair of parentheses \((\sqrt{c} + d)\) with each term in the second pair of parentheses \((3 + \sqrt{5})\). This process ensures that each term is accounted for and helps to break down complex expressions into simpler parts.
By expanding using the distributive property, we get two stages:

• Multiplying \(\sqrt{c}\) by each term, giving us \(\sqrt{c} \cdot 3\) and \(\sqrt{c} \cdot \sqrt{5}\).
• Multiplying \(d\) by each term, resulting in \(d \cdot 3\) and \(d \cdot \sqrt{5}\).

When you combine these products together, it helps form a complete expression that comprises all possible combinations of multiplied terms.

Radical expressions involve roots, such as square roots \(\sqrt{a}\). Working with radicals requires careful attention, especially during multiplication.
To multiply two radical expressions, like \(\sqrt{c}\) and \(\sqrt{5}\), you can use the property \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\). Applying this property in our solution gives us \(\sqrt{5c}\). Note that radicals behave predictably when multiplied, although they might seem intimidating at first glance.
It's important to remember, radical expressions can often be part of simplified results that retain their original values but look different. Simplifying them correctly is crucial in problem-solving. Also, ensure that you consider any additional constraints based on the context of the problem, such as the requirement for variables inside the radical to remain non-negative if you're working within the real number system.

Like terms in algebra refer to terms that have the same variable raised to the same power. Only like terms can be combined or simplified.
In the context of the exercise, while simplifying the expression, we identify terms that can be added together. These are terms with the same radical (such as \(\sqrt{c}\) and no radical (such as \(3d\)), which stand by themselves because no other matching terms are present.
For instance, \(3\sqrt{c}\) doesn’t have another similar term, and neither do \(\sqrt{5c}\), \(3d\), or \(d\sqrt{5}\). Each is unique in its nature due to differing coefficients or different radical forms. Understanding the properties of like terms helps ensure correct algebraic manipulation and makes simplification possible. Looking for these patterns advances the simplification process significantly and helps in reaching a final simplified expression.