Sure, let's simplify and solve these expressions step-by-step.
### Part d:
We have the expression [tex]\(2 \sqrt{2} - \sqrt{8} + 3 \sqrt{2}\)[/tex].
1. First, simplify [tex]\(\sqrt{8}\)[/tex]. We know that [tex]\(\sqrt{8} = \sqrt{4 \cdot 2} = 2 \sqrt{2}\)[/tex].
2. Substitute [tex]\(\sqrt{8}\)[/tex] with [tex]\(2 \sqrt{2}\)[/tex]:
[tex]\[
2 \sqrt{2} - 2 \sqrt{2} + 3 \sqrt{2}
\][/tex]
3. Combine the like terms:
[tex]\[
(2 \sqrt{2} - 2 \sqrt{2}) + 3 \sqrt{2} = 0 + 3 \sqrt{2} = 3 \sqrt{2}
\][/tex]
4. Thus the expression simplifies to [tex]\(3 \sqrt{2}\)[/tex].
The numeric value of [tex]\(3 \sqrt{2}\)[/tex] is approximately [tex]\(4.242640687119286\)[/tex].
### Part e:
We have the expression [tex]\(7 \sqrt{54} - 3 \sqrt{18} + \sqrt{24} - \frac{3}{5} \sqrt{50} - \sqrt{6}\)[/tex].
1. Simplify each term individually:
- [tex]\(7 \sqrt{54}\)[/tex]: [tex]\(\sqrt{54} = \sqrt{9 \cdot 6} = 3 \sqrt{6}\)[/tex]. Therefore, [tex]\(7 \sqrt{54} = 7 \cdot 3 \sqrt{6} = 21 \sqrt{6}\)[/tex].
- [tex]\(-3 \sqrt{18}\)[/tex]: [tex]\(\sqrt{18} = \sqrt{9 \cdot 2} = 3 \sqrt{2}\)[/tex]. Therefore, [tex]\(-3 \sqrt{18} = -3 \cdot 3 \sqrt{2} = -9 \sqrt{2}\)[/tex].
- [tex]\(\sqrt{24}\)[/tex]: [tex]\(\sqrt{24} = \sqrt{4 \cdot 6} = 2 \sqrt{6}\)[/tex].
- [tex]\(-\frac{3}{5} \sqrt{50}\)[/tex]: [tex]\(\sqrt{50} = \sqrt{25 \cdot 2} = 5 \sqrt{2}\)[/tex]. Therefore, [tex]\(-\frac{3}{5} \sqrt{50} = -\frac{3}{5} \cdot 5 \sqrt{2} = -3 \sqrt{2}\)[/tex].
- [tex]\(-\sqrt{6}\)[/tex].
2. Combine all the simplified terms:
[tex]\[
21 \sqrt{6} - 9 \sqrt{2} + 2 \sqrt{6} - 3 \sqrt{2} - \sqrt{6}
\][/tex]
3. Group like terms:
[tex]\[
(21 \sqrt{6} + 2 \sqrt{6} - \sqrt{6}) + (-9 \sqrt{2} - 3 \sqrt{2})
\][/tex]
[tex]\[
(21 \sqrt{6} + 2 \sqrt{6} - \sqrt{6}) = (21 + 2 - 1) \sqrt{6} = 22 \sqrt{6}
\][/tex]
[tex]\[
(-9 \sqrt{2} - 3 \sqrt{2}) = (-9 - 3) \sqrt{2} = -12 \sqrt{2}
\][/tex]
4. Therefore, the simplified expression is:
[tex]\[
22 \sqrt{6} - 12 \sqrt{2}
\][/tex]
The numeric value of this is approximately [tex]\(36.91821159275277\)[/tex].
So, to summarize, the solutions are:
- For part d: [tex]\(4.242640687119286\)[/tex]
- For part e: [tex]\(36.91821159275277\)[/tex]
