To solve the expression [tex]\(\sqrt{125} + 2\sqrt{27} - 5\sqrt{5} - \sqrt{3}\)[/tex], follow these steps:
1. Simplify the square roots:
- [tex]\(\sqrt{125}\)[/tex]: We can break this down as [tex]\(\sqrt{125} = \sqrt{5^2 \times 5} = 5\sqrt{5}\)[/tex].
- [tex]\(\sqrt{27}\)[/tex]: We can break this down as [tex]\(\sqrt{27} = \sqrt{3^2 \times 3} = 3\sqrt{3}\)[/tex].
2. Substitute the simplified terms back into the expression:
- The expression now looks like this:
[tex]\[
5\sqrt{5} + 2 \cdot 3\sqrt{3} - 5\sqrt{5} - \sqrt{3}
\][/tex]
3. Simplify the multiplication and addition within the terms:
- [tex]\(2 \cdot 3\sqrt{3} = 6\sqrt{3}\)[/tex]
4. Combine like terms:
- Combine the terms involving [tex]\(\sqrt{5}\)[/tex]:
[tex]\[
5\sqrt{5} - 5\sqrt{5} = 0
\][/tex]
- Combine the terms involving [tex]\(\sqrt{3}\)[/tex]:
[tex]\[
6\sqrt{3} - \sqrt{3} = 5\sqrt{3}
\][/tex]
5. Consolidate the expression:
- Since the [tex]\(\sqrt{5}\)[/tex] terms cancel each other out, we are left with:
[tex]\[
5\sqrt{3}
\][/tex]
Now, if we convert this result numerically:
[tex]\[
5\sqrt{3} \approx 5 \times 1.732 \approx 8.66
\][/tex]
Therefore, the simplified result of the given expression [tex]\(\sqrt{125} + 2\sqrt{27} - 5\sqrt{5} - \sqrt{3}\)[/tex] is approximately [tex]\(8.660254037844387\)[/tex].
