To simplify the given expression, we need to follow a step-by-step process. Let's look at the expression piece by piece and combine like terms.
The expression is:
[tex]\[
\sqrt{8} + 3\sqrt{2} + \sqrt{32} + 3\sqrt{8} + 3\sqrt{2} + 5\sqrt{42} + 9\sqrt{2} + 5\sqrt{2} + \sqrt{32}
\][/tex]
First, we'll simplify each term where possible:
1. [tex]\(\sqrt{8}\)[/tex] can be simplified:
[tex]\[
\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}
\][/tex]
2. [tex]\(\sqrt{32}\)[/tex] can be simplified:
[tex]\[
\sqrt{32} = \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4\sqrt{2}
\][/tex]
3. [tex]\(\sqrt{8}\)[/tex] appears again, simplified the same way:
[tex]\[
3\sqrt{8} = 3 \times 2\sqrt{2} = 6\sqrt{2}
\][/tex]
4. For the remaining terms, gather like terms:
[tex]\[
3\sqrt{2}, \quad 3\sqrt{2}, \quad 9\sqrt{2}, \quad 5\sqrt{2}
\][/tex]
Now let's gather all the like terms together:
- Combining [tex]\(\sqrt{2}\)[/tex] terms:
[tex]\[
2\sqrt{2} + 3\sqrt{2} + 4\sqrt{2} + 6\sqrt{2} + 3\sqrt{2} + 9\sqrt{2} + 5\sqrt{2} + 4\sqrt{2}
\][/tex]
First, sum them:
[tex]\[
2\sqrt{2} + 3\sqrt{2} + 4\sqrt{2} + 6\sqrt{2} + 3\sqrt{2} + 9\sqrt{2} + 5\sqrt{2} + 4\sqrt{2}
\][/tex]
[tex]\[
= (2 + 3 + 4 + 6 + 3 + 9 + 5 + 4)\sqrt{2}
\][/tex]
[tex]\[
= 36\sqrt{2}
\][/tex]
- Combining [tex]\(\sqrt{42}\)[/tex] terms:
[tex]\[
5\sqrt{42}
\][/tex]
Therefore, the expression in its simplest form is:
[tex]\[
5\sqrt{42} + 36\sqrt{2}
\][/tex]
Thus, the simplified sum of the given expression is:
[tex]\[
5\sqrt{42} + 36\sqrt{2}
\][/tex]
