Sure! Let's go through each part of the problem step-by-step.
### 1. Simplify and Perform Operations: [tex]\(7 \sqrt{20} - \sqrt{80} + 3 \sqrt{45}\)[/tex]
#### Simplifying Each Term:
1. [tex]\(7 \sqrt{20}\)[/tex]
- [tex]\(\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}\)[/tex]
- Therefore, [tex]\(7 \sqrt{20} = 7 \times 2\sqrt{5} = 14\sqrt{5}\)[/tex]
2. [tex]\(\sqrt{80}\)[/tex]
- [tex]\(\sqrt{80} = \sqrt{16 \times 5} = 4\sqrt{5}\)[/tex]
3. [tex]\(3 \sqrt{45}\)[/tex]
- [tex]\(\sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5}\)[/tex]
- Therefore, [tex]\(3 \sqrt{45} = 3 \times 3\sqrt{5} = 9\sqrt{5}\)[/tex]
#### Combine Simplified Terms:
- [tex]\(7 \sqrt{20} - \sqrt{80} + 3 \sqrt{45} = 14\sqrt{5} - 4\sqrt{5} + 9\sqrt{5}\)[/tex]
- Combine like terms: [tex]\( (14 - 4 + 9) \sqrt{5} = 19\sqrt{5}\)[/tex]
So, the result of [tex]\(7 \sqrt{20} - \sqrt{80} + 3 \sqrt{45}\)[/tex] is [tex]\(19 \sqrt{5}\)[/tex].
Numerically, [tex]\(7 \sqrt{20} \approx 31.30\)[/tex], [tex]\(\sqrt{80} \approx 8.94\)[/tex], and [tex]\(3 \sqrt{45} \approx 20.12\)[/tex].
Thus, [tex]\(31.30 - 8.94 + 20.12 \approx 42.48\)[/tex].
### 2. Simplify and Evaluate: [tex]\(10 + 9\sqrt{5}\)[/tex]
- This expression is already in its simplest form.
- Numerically, [tex]\(9 \sqrt{5} \approx 20.12\)[/tex], so [tex]\(10 + 9 \sqrt{5} \approx 10 + 20.12 \approx 30.12\)[/tex].
### 3. Simplify and Evaluate: [tex]\(19 \sqrt{5}\)[/tex]
- This expression is already in its simplest form.
- Numerically, [tex]\(19 \sqrt{5} \approx 42.48\)[/tex].
### 4. Simplify and Evaluate: [tex]\(\sqrt{195}\)[/tex]
- [tex]\(\sqrt{195}\)[/tex] can also be expressed in its decimal form.
- Numerically, [tex]\(\sqrt{195} \approx 13.96\)[/tex].
### 5. Simplify and Evaluate: [tex]\(\sqrt{5}\)[/tex]
- [tex]\(\sqrt{5}\)[/tex] is already in its simplest form.
- Numerically, [tex]\(\sqrt{5} \approx 2.24\)[/tex].
### Summary of Results:
1. [tex]\(7 \sqrt{20} - \sqrt{80} + 3 \sqrt{45} = 19\sqrt{5} \approx 42.48\)[/tex]
2. [tex]\(10 + 9\sqrt{5} \approx 30.12\)[/tex]
3. [tex]\(19\sqrt{5} \approx 42.48\)[/tex]
4. [tex]\(\sqrt{195} \approx 13.96\)[/tex]
5. [tex]\(\sqrt{5} \approx 2.24\)[/tex]
These are the simplified forms and numerical approximations for each part of the given question.
