Verified Answer :

  • Multiply the binomials using the distributive property (FOIL): \((3\sqrt{3}-5)(2\sqrt{6}-4) = (3\sqrt{3})(2\sqrt{6}) - (3\sqrt{3})(4) - (5)(2\sqrt{6}) + (5)(4)\).
  • Simplify each term: \(18\sqrt{2} - 12\sqrt{3} - 10\sqrt{6} + 20\).
  • Arrange the terms for clarity to reach the simplest form: \(\boxed{20 + 18\sqrt{2} - 12\sqrt{3} - 10\sqrt{6}}\).

Explanation

  1. Expansion of the Expression

We are given the expression \((3\sqrt{3}-5)(2\sqrt{6}-4)\). To simplify this, we apply the distributive property, often referred to as the FOIL method, which states that for any four terms, \((a-b)(c-d) = ac - ad - bc + bd\).
Let \(a = 3\sqrt{3}\), \(b = 5\), \(c = 2\sqrt{6}\), and \(d = 4\).
The expansion is:
\[(3\sqrt{3} \cdot 2\sqrt{6}) - (3\sqrt{3} \cdot 4) - (5 \cdot 2\sqrt{6}) + (5 \cdot 4)\]

  1. Simplification of Terms

Now we calculate each product individually:

  1. First term: \(3\sqrt{3} \cdot 2\sqrt{6} = 6\sqrt{3 \cdot 6} = 6\sqrt{18}\). We know \(\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}\). So, \(6\sqrt{18} = 6 \cdot 3\sqrt{2} = 18\sqrt{2}\).
  2. Second term: \(3\sqrt{3} \cdot (-4) = -12\sqrt{3}\).
  3. Third term: \(-5 \cdot 2\sqrt{6} = -10\sqrt{6}\).
  4. Fourth term: \(-5 \cdot (-4) = 20\).

Combining these parts, we get the expression: \(18\sqrt{2} - 12\sqrt{3} - 10\sqrt{6} + 20\).

  1. Final Simplified Form

The simplified expression is \(20 + 18\sqrt{2} - 12\sqrt{3} - 10\sqrt{6}\). Since \(\sqrt{2}\), \(\sqrt{3}\), and \(\sqrt{6}\) are linearly independent square roots, no further simplification is possible. The final result is \(\boxed{20 + 18\sqrt{2} - 12\sqrt{3} - 10\sqrt{6}}\).

Examples

Simplifying radical expressions is a foundational skill in geometry and physics. For example, when calculating the area of a non-rectangular shape divided into smaller triangles, you might encounter side lengths involving roots. Combining these expressions using algebraic rules allows you to find a single, precise value for the total area, which is useful in architecture or landscaping to ensure correct material quantities.

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