Verified Answer :

  • Distribute the terms using the FOIL method: \((6 \times 3) + (6 \times \sqrt{3}) - (\sqrt{3} \times 3) - (\sqrt{3} \times \sqrt{3})\).
  • Evaluate each multiplication to get \(18 + 6\sqrt{3} - 3\sqrt{3} - 3\).
  • Simplify the expression by grouping rational and irrational parts: \((18-3) + (6\sqrt{3}-3\sqrt{3})\).
  • The final simplified result is \(\boxed{15+3\sqrt{3}}\).

Explanation

  1. Understanding the Problem

The goal is to simplify the product of two binomials involving square roots: \((6-\sqrt{3})(3+\sqrt{3})\). This expression is in the form of \((a-b)(c+d)\). We can simplify it by using the distributive property, commonly known as the FOIL method (First, Outer, Inner, Last), which ensures each term in the first binomial is multiplied by each term in the second binomial.

  1. Expanding the Expression

Using the FOIL method, we multiply the terms as follows:

  • First: \(6 \times 3 = 18\)
  • Outer: \(6 \times \sqrt{3} = 6\sqrt{3}\)
  • Inner: \(-\sqrt{3} \times 3 = -3\sqrt{3}\)
  • Last: \(-\sqrt{3} \times \sqrt{3} = -(\sqrt{3})^2 = -3\)

Adding these results together, we get:
\(18 + 6\sqrt{3} - 3\sqrt{3} - 3\)

  1. Combining Like Terms and Final Result

Now, we combine the rational parts and the irrational parts involving \(\sqrt{3}\):

  • Combine the rational numbers: \(18 - 3 = 15\)
  • Combine the radical terms: \(6\sqrt{3} - 3\sqrt{3} = 3\sqrt{3}\)

Combining these simplified results yields the expression \(15 + 3\sqrt{3}\). This matches the fourth option provided.

Examples

In real-world applications, such as calculating the area of a rectangle with sides \((6-\sqrt{3})\) meters and \((3+\sqrt{3})\) meters, you perform this exact type of multiplication to find the total area. Simplifying such radical expressions is fundamental in geometry, construction, and engineering when dealing with precise physical dimensions involving square roots.

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