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The distributive property is a fundamental concept in algebra that allows us to multiply a single term by each term within a parenthesis. It states that for any numbers \( a, b, \) and \( c \), the expression \( a(b + c) \) is equivalent to \( ab + ac \). In our example, we use the distributive property to expand the brackets in the radical expression \((2 \sqrt{z} - \sqrt{3})(\sqrt{z} - \sqrt{5})\). By applying this property, we multiply each term in the first binomial by every term in the second binomial:

\[ 2 \sqrt{z} \cdot \sqrt{z} + 2 \sqrt{z} \cdot - \sqrt{5} - \sqrt{3} \cdot \sqrt{z} - \sqrt{3} \cdot - \sqrt{5} \]
This step ensures that no term is left out, laying the groundwork for further simplification.

The FOIL method is a specific application of the distributive property used for multiplying two binomials. FOIL stands for First, Outer, Inner, and Last, representing the order in which you multiply the terms:

- First: Multiply the first terms from each binomial: \( 2 \sqrt{z} \cdot \sqrt{z} \)
- Outer: Multiply the outer terms: \( 2 \sqrt{z} \cdot - \sqrt{5} \)
- Inner: Multiply the inner terms: \( - \sqrt{3} \cdot \sqrt{z} \)
- Last: Multiply the last terms: \( - \sqrt{3} \cdot - \sqrt{5} \)

This method helps ensure all components are considered, resulting in expressions like:
\[ (2 \sqrt{z} \cdot \sqrt{z}) + (2 \sqrt{z} \cdot - \sqrt{5}) - (\sqrt{3} \cdot \sqrt{z}) - (\sqrt{3} \cdot - \sqrt{5}) \]

Combining like terms involves adding or subtracting terms that have the same variable and exponent. In the simplified expression
\( 2z - 2 \sqrt{5z} - \sqrt{3z} + \sqrt{15} \), each term must be inspected:

- \( 2z \) is a single term with no radicals
- Terms like \( -2 \sqrt{5z} \) and \( - \sqrt{3z} \) are mixed radicals and can't be combined directly but can be grouped together for clarity
- \( \sqrt{15} \) remains as it is

Because there are no like terms, this expression --> \( 2z - 2 \sqrt{5z} - \sqrt{3z} + \sqrt{15} \) <-- is left in its simplified form.

The multiplication of radicals follows a straightforward rule: Multiply the numbers outside the radicals and then multiply the numbers inside the radicals. For instance:

\( 2 \sqrt{z} \cdot \sqrt{z} \) results in \( 2 \cdot z \) which is\( 2z \)
Another example is \( - \sqrt{3} \cdot \sqrt{z} = - \sqrt{3z} \)

Remember, when you end up with a product under the radical that can be simplified further (like \( \sqrt{16} \) turning into \( 4 \)), always perform that simplification. Knowing these rules ensures you can handle products involving radicals skillfully.