Question

what is the simplified form of the following expression?
$2\sqrt{18} + 3\sqrt{2} + \sqrt{162}$
options:
$6\sqrt{2}$
$36\sqrt{2}$
$18\sqrt{2}$
$30\sqrt{2}$

Explanation:

Step1: Simplify \(2\sqrt{18}\)

We know that \(18 = 9\times2\), so \(\sqrt{18}=\sqrt{9\times2}=\sqrt{9}\times\sqrt{2}=3\sqrt{2}\). Then \(2\sqrt{18}=2\times3\sqrt{2}=6\sqrt{2}\).

Step2: Simplify \(\sqrt{162}\)

Since \(162 = 81\times2\), \(\sqrt{162}=\sqrt{81\times2}=\sqrt{81}\times\sqrt{2}=9\sqrt{2}\).

Step3: Combine like terms

Now the expression \(2\sqrt{18}+3\sqrt{2}+\sqrt{162}\) becomes \(6\sqrt{2}+3\sqrt{2}+9\sqrt{2}\). Combining the coefficients: \(6 + 3+9 = 18\), so the result is \(18\sqrt{2}\).

Answer:

\(18\sqrt{2}\)