- Simplify the radical terms by factoring out perfect squares: \(\sqrt{18} = 3\sqrt{2}\) and \(\sqrt{162} = 9\sqrt{2}\).
- Substitute these back into the expression: \(2(3\sqrt{2}) + 3\sqrt{2} + 9\sqrt{2}\).
- Combine the like terms by adding the coefficients: \((6 + 3 + 9)\sqrt{2} = 18\sqrt{2}\).
- The simplified form is \(\boxed{18\sqrt{2}}\).
Explanation
- Analysis and Simplification Strategy
We are tasked with simplifying the expression \(2\sqrt{18} + 3\sqrt{2} + \sqrt{162}\). To simplify expressions involving square roots, we look for factors of the radicands (the numbers inside the square root) that are perfect squares. A perfect square factor \(k^2\) can be pulled out of the square root as \(k\), because \(\sqrt{k^2 \cdot m} = k\sqrt{m}\). Once all terms are expressed in terms of the same radical (in this case, \(\sqrt{2}\)), we can combine them by adding their coefficients.
- Simplifying \(\sqrt{18}\)
First, we simplify \(\sqrt{18}\). Since \(18 = 9 \times 2\) and 9 is a perfect square (\(3^2\)), we have:
\[\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2}\]
Substituting this into the first term of our expression, we get \(2\sqrt{18} = 2(3\sqrt{2}) = 6\sqrt{2}\).
- Simplifying \(\sqrt{162}\)
Next, we simplify \(\sqrt{162}\). We look for a perfect square factor of 162. Since \(162 = 81 \times 2\) and 81 is a perfect square (\(9^2\)), we have:
\[\sqrt{162} = \sqrt{81 \times 2} = \sqrt{81} \times \sqrt{2} = 9\sqrt{2}\]
- Combining Terms and Final Result
Now, we substitute the simplified forms back into the original expression:
\[2\sqrt{18} + 3\sqrt{2} + \sqrt{162} = (6\sqrt{2}) + 3\sqrt{2} + (9\sqrt{2})\]
Since all terms are now multiples of \(\sqrt{2}\), we can add their coefficients:
\[(6 + 3 + 9)\sqrt{2} = 18\sqrt{2}\]
Thus, the simplified form of the expression is \(18\sqrt{2}\). Comparing this to the given options, it matches the second option. The final result is \(\boxed{18 \sqrt{2}}\).
Examples
Simplifying radicals is a fundamental skill in algebra that appears frequently in engineering and physics. For example, if you are calculating the total length of a wire frame for a sculpture consisting of several segments with lengths involving square roots, you would use this exact method to combine the lengths into a single, manageable value, ensuring accuracy in material procurement. It is also used when simplifying complex equations in electricity or structural analysis, where radical terms often arise from geometric formulas involving the Pythagorean theorem.