Verified Answer :

  • Regroup the terms by multiplying the integer coefficients and the radicals separately: \((4 \cdot 6) \cdot (\sqrt{5} \cdot \sqrt{8})\).
  • Multiply the coefficients to get \(24\) and combine the radicals to get \(\sqrt{40}\).
  • Simplify \(\sqrt{40}\) into \(2 \sqrt{10}\) by factoring out the perfect square \(4\).
  • Multiply the results to reach the final simplified form: \(\boxed{48 \sqrt{10}}\).

Explanation

  1. Analyzing the Expression

The goal is to simplify the expression \(4 \sqrt{5} \cdot 6 \sqrt{8}\). We start by identifying the components: the integer coefficients \(4\) and \(6\), and the radicands (numbers inside the square roots) \(5\) and \(8\). According to the properties of multiplication, we can regroup the terms to multiply the coefficients and the radicals separately.

  1. Multiplying Coefficients and Radicals

Using the commutative property of multiplication, we rearrange the expression as follows:
\[4 \sqrt{5} \cdot 6 \sqrt{8} = (4 \cdot 6) \cdot (\sqrt{5} \cdot \sqrt{8})\]
First, multiply the coefficients:
\[4 \cdot 6 = 24\]
Next, multiply the radical terms using the product rule for radicals, \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\):
\[\sqrt{5} \cdot \sqrt{8} = \sqrt{5 \cdot 8} = \sqrt{40}\]
Combining these, we get:
\[24 \sqrt{40}\]

  1. Simplifying the Radical and Finalizing the Product

Now, we simplify the square root of \(40\). We look for the largest perfect square that is a factor of \(40\). Since \(40 = 4 \cdot 10\) and \(4\) is a perfect square, we can write:
\[\sqrt{40} = \sqrt{4 \cdot 10} = \sqrt{4} \cdot \sqrt{10}\]
Knowing that \(\sqrt{4} = 2\), the expression becomes:
\[\sqrt{40} = 2 \sqrt{10}\]
Finally, we substitute this back into our expression:
\[24 \cdot 2 \sqrt{10} = 48 \sqrt{10}\]
Thus, the simplified form is \(\boxed{48 \sqrt{10}}\).

Examples

In civil engineering and construction, radical expressions frequently appear when calculating distances using the Pythagorean theorem or areas of shapes involving non-perfect square dimensions. For instance, if you need to calculate the area of a rectangle with sides of length \(4 \sqrt{5}\) meters and \(6 \sqrt{8}\) meters, the resulting area of \(48 \sqrt{10}\) square meters allows for a more precise estimation when working with building materials, as \(\sqrt{10}\) can be approximated to about \(3.16\) to determine the required amount of floor surface or material. Simplifying radicals helps engineers maintain precision throughout their calculations before converting to decimal values at the final stage.

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