To simplify the given expressions and find the sum in its simplest form, let's break down each term step by step.
1. Simplify each term individually:
- [tex]\(\sqrt{8}\)[/tex]:
[tex]\[
\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}
\][/tex]
Hence, [tex]\(\sqrt{8} = 2\sqrt{2}\)[/tex].
- [tex]\(3\sqrt{2}\)[/tex]:
This term is already in its simplest form.
- [tex]\(\sqrt{32}\)[/tex]:
[tex]\[
\sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2}
\][/tex]
Hence, [tex]\(\sqrt{32} = 4\sqrt{2}\)[/tex].
- [tex]\(3\sqrt{8}\)[/tex]:
Using the simplified form of [tex]\(\sqrt{8}\)[/tex]:
[tex]\[
3\sqrt{8} = 3 \times 2\sqrt{2} = 6\sqrt{2}
\][/tex]
- [tex]\(3\sqrt{2}\)[/tex]:
This term is already in its simplest form.
- [tex]\(5\sqrt{42}\)[/tex]:
This term is already in its simplest form and cannot be simplified further.
- [tex]\(9\sqrt{2}\)[/tex]:
This term is already in its simplest form.
- [tex]\(5\sqrt{2} + \sqrt{32}\)[/tex]:
Using the simplified form of [tex]\(\sqrt{32}\)[/tex]:
[tex]\[
5\sqrt{2} + \sqrt{32} = 5\sqrt{2} + 4\sqrt{2} = 9\sqrt{2}
\][/tex]
2. Combine the simplified terms step by step:
- First part:
[tex]\[
\sqrt{8} + 3\sqrt{2} + \sqrt{32} = 2\sqrt{2} + 3\sqrt{2} + 4\sqrt{2}
\][/tex]
Combine the coefficients:
[tex]\[
2\sqrt{2} + 3\sqrt{2} + 4\sqrt{2} = (2 + 3 + 4)\sqrt{2} = 9\sqrt{2}
\][/tex]
- Second part:
[tex]\[
3\sqrt{8} + 3\sqrt{2} = 6\sqrt{2} + 3\sqrt{2}
\][/tex]
Combine the coefficients:
[tex]\[
6\sqrt{2} + 3\sqrt{2} = (6 + 3)\sqrt{2} = 9\sqrt{2}
\][/tex]
- Third part remains as:
[tex]\[
5\sqrt{42}
\][/tex]
- Fourth part remains as:
[tex]\[
9\sqrt{2}
\][/tex]
- Fifth part:
[tex]\[
5\sqrt{2} + \sqrt{32} = 5\sqrt{2} + 4\sqrt{2} = 9\sqrt{2}
\][/tex]
3. Combine all the parts:
- Simplify the sum of all the combined parts from above:
[tex]\[
9\sqrt{2} + 9\sqrt{2} + 5\sqrt{42} + 9\sqrt{2} + 9\sqrt{2}
\][/tex]
Combine all similar terms:
[tex]\[
(9 + 9 + 9 + 9)\sqrt{2} + 5\sqrt{42} = 36\sqrt{2} + 5\sqrt{42}
\][/tex]
Therefore, the sum of the given terms in simplest form is:
[tex]\[
36\sqrt{2} + 5\sqrt{42}
\][/tex]
And using the numerical results obtained from simplifying the exact numerical calculation:
[tex]\[
12.727922061357855
\][/tex]
Hence, the detailed step-by-step simplification process for the sum is shown above.
