Verified Answer :

  • Expand the expression \((7-\sqrt{2})(8+\sqrt{2})\) using the FOIL method: \((7 \times 8) + (7 \times \sqrt{2}) - (\sqrt{2} \times 8) - (\sqrt{2} \times \sqrt{2})\).
  • Evaluate each part: \(56 + 7\sqrt{2} - 8\sqrt{2} - 2\).
  • Combine constants and terms with \(\sqrt{2}\) to get \(54 - \sqrt{2}\).
  • The final simplified expression is \(\boxed{54-\sqrt{2}}\).

Explanation

  1. Data and Problem Analysis

We are given the expression \((7-\sqrt{2})(8+\sqrt{2})\). This expression is a product of two binomials. To simplify it, we can use the distributive property, often referred to as the FOIL method (First, Outer, Inner, Last). Our goal is to expand the product and simplify the result to match one of the given choices. The values are \(a=7, b=\sqrt{2}, c=8,\) and \(d=\sqrt{2}\). The structure is \((a-b)(c+d) = ac + ad - bc - bd\).

  1. Expanding the Expression

We apply the distributive property (FOIL method) to the expression:
\[ (7-\sqrt{2})(8+\sqrt{2}) = 7 \cdot 8 + 7 \cdot \sqrt{2} - \sqrt{2} \cdot 8 - \sqrt{2} \cdot \sqrt{2} \]
Calculating each product:

  • First: \(7 \cdot 8 = 56\)
  • Outer: \(7 \cdot \sqrt{2} = 7\sqrt{2}\)
  • Inner: \(-\sqrt{2} \cdot 8 = -8\sqrt{2}\)
  • Last: \(-\sqrt{2} \cdot \sqrt{2} = -(\sqrt{2})^2 = -2\)

Substituting these values back into the expression, we get:
\[ 56 + 7\sqrt{2} - 8\sqrt{2} - 2 \]

  1. Simplification and Final Result

Now we group the constant terms and the terms involving \(\sqrt{2}\):

  • Constants: \(56 - 2 = 54\)
  • Terms with \(\sqrt{2}\): \(7\sqrt{2} - 8\sqrt{2} = (7-8)\sqrt{2} = -1\sqrt{2} = -\sqrt{2}\)

Combining these gives:
\[ 54 - \sqrt{2} \]
Comparing this result to the provided options, we find that it matches the third option.

Examples

This algebraic simplification is useful in various engineering and scientific applications where square roots appear in dimensions or physical constants. For example, if you are calculating the area of a rectangle with sides \((7-\sqrt{2})\) units and \((8+\sqrt{2})\) units, the resulting area in square units would be \((7-\sqrt{2})(8+\sqrt{2}) = 54-\sqrt{2} \approx 52.585\). Understanding how to handle these radicals correctly ensures accuracy in construction, physics simulations, and advanced mathematics.

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