Verified Answer :

  • Apply commutative and associative properties of multiplication to group the integers and radicals separately: \( (2 \cdot 3) \cdot (\sqrt{6} \cdot \sqrt{6}) \).
  • Calculate the product of the integer coefficients: \(2 \cdot 3 = 6\).
  • Calculate the product of the square roots: \(\sqrt{6} \cdot \sqrt{6} = 6\).
  • Multiply the two results to find the final value: \(6 \cdot 6 = \boxed{36}\).

Explanation

  1. Analyzing the Expression

The expression to evaluate is \(2\sqrt{6} \cdot 3\sqrt{6}\). This expression involves the multiplication of two terms, each consisting of an integer coefficient and a square root. To simplify this, we can rely on the associative and commutative properties of multiplication, which allow us to rearrange the factors:
\[ 2\sqrt{6} \cdot 3\sqrt{6} = (2 \cdot 3) \cdot (\sqrt{6} \cdot \sqrt{6}) \]
This structure helps us focus on multiplying the integers and the radicals separately.

  1. Evaluating and Calculating the Final Result

Next, we evaluate the product of the integer coefficients and the product of the radicals separately:

  1. Integer part: \(2 \cdot 3 = 6\)
  2. Radical part: \(\sqrt{6} \cdot \sqrt{6} = (\sqrt{6})^2 = 6\)

By multiplying these two results together, we obtain the final product:
\[ 6 \cdot 6 = 36 \]
Thus, the simplification leads us to the final result of 36.

Examples

When measuring the dimensions of a rectangular plot of land, if the length is \(2\sqrt{6}\) meters and the width is \(3\sqrt{6}\) meters, the area of the plot would be the product of these dimensions. By simplifying the expression \(2\sqrt{6} \cdot 3\sqrt{6}\), we find the area to be exactly \(36\) square meters, which is a much more practical number for planning landscaping or construction.

Related Questions