To simplify the expression [tex]\(\sqrt{80} - 2 \sqrt{45}\)[/tex], follow these steps:
1. Simplify [tex]\(\sqrt{80}\)[/tex]:
- [tex]\(\sqrt{80}\)[/tex] can be broken down into [tex]\(\sqrt{16 \cdot 5}\)[/tex].
- Use the property [tex]\(\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}\)[/tex]:
[tex]\[
\sqrt{80} = \sqrt{16 \cdot 5} = \sqrt{16} \cdot \sqrt{5}
\][/tex]
- Since [tex]\(\sqrt{16} = 4\)[/tex]:
[tex]\[
\sqrt{80} = 4 \cdot \sqrt{5}
\][/tex]
Numerically, [tex]\(\sqrt{80} \approx 8.94427190999916\)[/tex].
2. Simplify [tex]\(2 \sqrt{45}\)[/tex]:
- [tex]\(\sqrt{45}\)[/tex] can be broken down into [tex]\(\sqrt{9 \cdot 5}\)[/tex].
- Use the property [tex]\(\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}\)[/tex]:
[tex]\[
\sqrt{45} = \sqrt{9 \cdot 5} = \sqrt{9} \cdot \sqrt{5}
\][/tex]
- Since [tex]\(\sqrt{9} = 3\)[/tex]:
[tex]\[
\sqrt{45} = 3 \cdot \sqrt{5}
\][/tex]
- Multiply by 2:
[tex]\[
2 \sqrt{45} = 2 \cdot (3 \cdot \sqrt{5}) = 6 \cdot \sqrt{5}
\][/tex]
Numerically, [tex]\(2 \sqrt{45} \approx 13.416407864998739\)[/tex].
3. Combine the simplified parts:
- The expression becomes:
[tex]\[
\sqrt{80} - 2 \sqrt{45} = 4 \cdot \sqrt{5} - 6 \cdot \sqrt{5}
\][/tex]
- Since [tex]\(4 \cdot \sqrt{5}\)[/tex] minus [tex]\(6 \cdot \sqrt{5}\)[/tex] is [tex]\((4 - 6) \cdot \sqrt{5}\)[/tex]:
[tex]\[
\sqrt{80} - 2 \sqrt{45} = -2 \cdot \sqrt{5}
\][/tex]
Numerically, this results in [tex]\(\approx -4.47213595499958\)[/tex].
So, the simplified expression is [tex]\(-2 \cdot \sqrt{5}\)[/tex].
The results you should enter are:
1. [tex]\(\sqrt{80} \approx 8.94427190999916\)[/tex]
2. [tex]\(2 \sqrt{45} \approx 13.416407864998739\)[/tex]
3. The final simplified result: [tex]\(-2 \cdot \sqrt{5} \approx -4.47213595499958\)[/tex]
