Question

what is the following sum in simplest form?
\\(\sqrt{8} + 3\sqrt{2} + \sqrt{32}\\)
\\(9\sqrt{2}\\) \\(3\sqrt{8} + 3\sqrt{2}\\) \\(5\sqrt{42}\\)
\\(5\sqrt{2} + \sqrt{32}\\)

Explanation:

Step1: Simplify \(\sqrt{8}\)

We know that \(8 = 4\times2\), so \(\sqrt{8}=\sqrt{4\times2}\). By the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (\(a\geq0,b\geq0\)), we have \(\sqrt{4\times2}=\sqrt{4}\times\sqrt{2}=2\sqrt{2}\).

Step2: Simplify \(\sqrt{32}\)

Since \(32 = 16\times2\), then \(\sqrt{32}=\sqrt{16\times2}\). Using the same property as above, \(\sqrt{16\times2}=\sqrt{16}\times\sqrt{2}=4\sqrt{2}\).

Step3: Substitute and Combine Like Terms

Now substitute the simplified forms back into the original expression: \(\sqrt{8}+3\sqrt{2}+\sqrt{32}=2\sqrt{2}+3\sqrt{2}+4\sqrt{2}\). Combining like terms (adding the coefficients of \(\sqrt{2}\)), we get \((2 + 3+4)\sqrt{2}=9\sqrt{2}\).

Answer:

\(9\sqrt{2}\)