Verified Answer :

  • Apply the distributive property (FOIL method) to expand the binomial product \((3\sqrt{3}-5)(2\sqrt{6}-4)\).
  • Simplify the product of radicals by calculating \((3\sqrt{3})(2\sqrt{6}) = 18\sqrt{2}\).
  • Combine all expanded terms into the final expression: \(20 + 18\sqrt{2} - 12\sqrt{3} - 10\sqrt{6}\).
  • The final simplified form is \(\boxed{20 + 18\sqrt{2} - 12\sqrt{3} - 10\sqrt{6}}\).

Explanation

  1. Analysis of the Expression

The expression to simplify is \((3\sqrt{3}-5)(2\sqrt{6}-4)\). This is a product of two binomials. We can use the distributive property of multiplication (often referred to as the FOIL method) to expand this expression. The FOIL method states that for two binomials \((a+b)(c+d)\), the product is \(ac + ad + bc + bd\).

  1. Expanding the Expression

Applying the distributive property, we multiply each term in the first binomial by each term in the second:

  • First terms: \((3\sqrt{3}) \times (2\sqrt{6}) = 6\sqrt{18} = 6\sqrt{9 \times 2} = 6 \times 3\sqrt{2} = 18\sqrt{2}\)
  • Outer terms: \((3\sqrt{3}) \times (-4) = -12\sqrt{3}\)
  • Inner terms: \((-5) \times (2\sqrt{6}) = -10\sqrt{6}\)
  • Last terms: \((-5) \times (-4) = 20\)
  1. Combining and Simplifying

Now, we combine all the terms obtained:
\[18\sqrt{2} - 12\sqrt{3} - 10\sqrt{6} + 20\]
Rearranging for standard form, we have \(20 + 18\sqrt{2} - 12\sqrt{3} - 10\sqrt{6}\). There are no like terms to combine, so this is the simplest form. The final answer is \(\boxed{20 + 18\sqrt{2} - 12\sqrt{3} - 10\sqrt{6}}\).

Examples

Calculating products involving irrational numbers like \(\sqrt{3}\) and \(\sqrt{6}\) often appears in geometry, such as when finding the area of complex shapes composed of triangles or rectangles where side lengths are expressed as radicals. For instance, if you are designing a window frame with dimensions determined by these radical expressions, multiplying them correctly is essential to determine the exact material surface area required for construction. Keeping the expression in its exact form (using radicals) ensures precision in measurements before converting to decimals.

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