- Simplify each individual radical by factoring out perfect squares: \(\sqrt{8} = 2\sqrt{2}\) and \(\sqrt{18} = 3\sqrt{2}\).
- Group the rational coefficients and radical terms: \((1 \cdot 2 \cdot 10 \cdot 3) \cdot (\sqrt{3} \cdot \sqrt{2} \cdot \sqrt{2} \cdot \sqrt{2})\).
- Multiply the coefficients to get \(60\), and simplify the radicals to get \(2\sqrt{6}\).
- Multiply these components to arrive at the simplest radical form: \(\boxed{120\sqrt{6}}\).
Explanation
- Simplifying the Individual Radical Terms
We are given the product of four terms involving square roots: \(\sqrt{3} \cdot 2\sqrt{2} \cdot 5\sqrt{8} \cdot \sqrt{18}\). To simplify this, we can first simplify each individual radical term by identifying any perfect square factors within the radicands (the numbers inside the square roots). Recall that \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\).
- \(\sqrt{3}\) is already in simplest form.
- \(2\sqrt{2}\) is already in simplest form.
- \(5\sqrt{8} = 5\sqrt{4 \cdot 2} = 5 \cdot 2\sqrt{2} = 10\sqrt{2}\).
- \(\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}\).
- Multiplying the Simplified Terms
Now that we have simplified each part, we substitute them back into the original expression:
\[\sqrt{3} \cdot (2\sqrt{2}) \cdot (10\sqrt{2}) \cdot (3\sqrt{2})\]
We can group the rational coefficients (the numbers outside the square roots) together and the radical terms together:
\[(1 \cdot 2 \cdot 10 \cdot 3) \cdot (\sqrt{3} \cdot \sqrt{2} \cdot \sqrt{2} \cdot \sqrt{2})\]
Calculate the product of the coefficients:
\[1 \cdot 2 \cdot 10 \cdot 3 = 60\]
Now, simplify the product of the square roots:
\[\sqrt{3} \cdot \sqrt{2} \cdot \sqrt{2} \cdot \sqrt{2} = \sqrt{3} \cdot (\sqrt{2})^3 = \sqrt{3} \cdot (2\sqrt{2}) = 2\sqrt{6}\]
- Final Calculation and Conclusion
Combine the result of the coefficients and the radicals:
\[60 \cdot (2\sqrt{6}) = 120\sqrt{6}\]
The product of the given expression is \(120\sqrt{6}\). Therefore, the final answer in simplest radical form is \(\boxed{120\sqrt{6}}\).
Examples
In architecture and engineering, simplifying radical expressions is essential when calculating lengths or areas derived from geometric properties. For instance, if you are designing a structure where the area involves multiplying various segments with root-based lengths, simplifying these expressions first ensures accuracy in material estimation and structural integrity. By simplifying complex radical products, you make the final numerical values easier to interpret and work with in real-world construction projects.