- Simplify each term: \(\sqrt{8} = 2\sqrt{2}\) and \(\sqrt{18} = 3\sqrt{2}\).
- Substitute the simplified terms: \(\sqrt{3} \cdot 2\sqrt{2} \cdot 10\sqrt{2} \cdot 3\sqrt{2}\).
- Multiply the coefficients: \(1 \cdot 2 \cdot 10 \cdot 3 = 60\).
- Multiply the radicals: \(\sqrt{3} \cdot \sqrt{2} \cdot \sqrt{2} \cdot \sqrt{2} = 2\sqrt{6}\), then compute \(60 \cdot 2\sqrt{6} = 120\sqrt{6}\).
- The final answer is \(\boxed{120\sqrt{6}}\).
Explanation
- Analysis of the Problem
We are asked to compute the product \(\sqrt{3} \cdot 2\sqrt{2} \cdot 5\sqrt{8} \cdot \sqrt{18}\) and express the result in its simplest radical form. This involves using properties of square roots, specifically \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\) and \(\sqrt{a^2 \cdot b} = a\sqrt{b}\) to simplify each term before multiplying.
- Simplifying Each Radical Term
First, we simplify each radical term individually:
- \(\sqrt{3}\) is already in simplest form.
- \(2\sqrt{2}\) is already in simplest form.
- For \(\sqrt{8}\), since \(8 = 4 \cdot 2\), we have \(\sqrt{8} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2}\). Thus, \(5\sqrt{8} = 5(2\sqrt{2}) = 10\sqrt{2}\).
- For \(\sqrt{18}\), since \(18 = 9 \cdot 2\), we have \(\sqrt{18} = \sqrt{9} \cdot \sqrt{2} = 3\sqrt{2}\).
- Multiplying the Terms
Now, we substitute these simplified expressions back into the original product:
\[\sqrt{3} \cdot 2\sqrt{2} \cdot 10\sqrt{2} \cdot 3\sqrt{2}\]
Next, we group the coefficients (the numbers outside the radicals) and the radicands (the values inside the radicals) to multiply them separately:
- Product of coefficients: \(1 \cdot 2 \cdot 10 \cdot 3 = 60\).
- Product of radicals: \(\sqrt{3} \cdot \sqrt{2} \cdot \sqrt{2} \cdot \sqrt{2} = \sqrt{3} \cdot (\sqrt{2})^3\).
Since \((\sqrt{2})^3 = \sqrt{2} \cdot \sqrt{2} \cdot \sqrt{2} = 2\sqrt{2}\), the product of the radicals is \(\sqrt{3} \cdot 2\sqrt{2} = 2\sqrt{6}\).
- Conclusion and Final Answer
Finally, we combine the product of the coefficients and the simplified radicals:
\[60 \cdot 2\sqrt{6} = 120\sqrt{6}\]
Since \(\sqrt{6}\) has no perfect square factors other than 1, the expression \(120\sqrt{6}\) is in its simplest radical form. Thus, the final answer is \(\boxed{120\sqrt{6}}\).
Examples
Simplifying radical expressions is a foundational skill in fields like engineering and physics. For example, when calculating the diagonal of a rectangular structure using the Pythagorean theorem, you often end up with square roots of products. If you are designing a frame where components have lengths like \(\sqrt{8}\) and \(\sqrt{18}\) meters, being able to simplify these allows you to calculate the total length or area quickly and accurately, ensuring your measurements are precise before construction.