Verified Answer :

  • Use the FOIL method to expand the expression \((3\sqrt{7} + 2\sqrt{3})(3\sqrt{7} + 6\sqrt{3})\).
  • Multiply the terms: \(63\), \(18\sqrt{21}\), \(6\sqrt{21}\), and \(36\).
  • Add the constants (\(63 + 36 = 99\)) and the radicals (\(18\sqrt{21} + 6\sqrt{21} = 24\sqrt{21}\)).
  • Combine for the final answer: \(\boxed{99 + 24\sqrt{21}}\).

Explanation

  1. Analyzing the Problem

We are asked to find the product of two binomials involving square roots: \((3\sqrt{7} + 2\sqrt{3})\) and \((3\sqrt{7} + 6\sqrt{3})\). To expand this expression, we use the distributive property, often referred to as the FOIL method (First, Outer, Inner, Last). This will give us the sum of four separate products.

  1. Expanding the Binomials

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

  • First terms: \((3\sqrt{7}) \times (3\sqrt{7}) = 9 \times 7 = 63\)
  • Outer terms: \((3\sqrt{7}) \times (6\sqrt{3}) = 18 \times \sqrt{7 \times 3} = 18\sqrt{21}\)
  • Inner terms: \((2\sqrt{3}) \times (3\sqrt{7}) = 6 \times \sqrt{3 \times 7} = 6\sqrt{21}\)
  • Last terms: \((2\sqrt{3}) \times (6\sqrt{3}) = 12 \times 3 = 36\)
  1. Simplifying and Finalizing the Result

Now we sum the four products calculated in the previous step:
\[63 + 18\sqrt{21} + 6\sqrt{21} + 36\]
We group the constant terms and the terms involving \(\sqrt{21}\):

  • Sum of constants: \(63 + 36 = 99\)
  • Sum of square root terms: \(18\sqrt{21} + 6\sqrt{21} = (18 + 6)\sqrt{21} = 24\sqrt{21}\)

Adding these together, the final simplified result is \(99 + 24\sqrt{21}\).

Examples

Expanding binomials with radicals is essential in many fields, including construction and architecture, where diagonal lengths or areas of triangular plots are calculated. For example, if you are designing a rectangular garden with sections of side lengths involving square roots, calculating the total area or total length requires similar distribution techniques to ensure accuracy in material estimation. Proficiency in this skill ensures that calculations for complex geometric structures remain precise.

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