High School

Which choice is equivalent to the expression below?

[tex]\sqrt{6} + 2\sqrt{3} + \sqrt{27} - \sqrt{12}[/tex]

Answer :

To simplify the expression [tex]\(\sqrt{6} + 2\sqrt{3} + \sqrt{27} - \sqrt{12}\)[/tex], follow these steps:

1. Simplify [tex]\(\sqrt{27}\)[/tex]:
- Notice that 27 can be written as [tex]\(9 \times 3\)[/tex].
- The square root of 27 is [tex]\(\sqrt{27} = \sqrt{9 \times 3}\)[/tex].
- Since [tex]\(\sqrt{9} = 3\)[/tex], this simplifies to [tex]\(3\sqrt{3}\)[/tex].

2. Simplify [tex]\(\sqrt{12}\)[/tex]:
- Notice that 12 can be written as [tex]\(4 \times 3\)[/tex].
- The square root of 12 is [tex]\(\sqrt{12} = \sqrt{4 \times 3}\)[/tex].
- Since [tex]\(\sqrt{4} = 2\)[/tex], this simplifies to [tex]\(2\sqrt{3}\)[/tex].

3. Substitute the simplified forms back into the expression:
- Replace [tex]\(\sqrt{27}\)[/tex] with [tex]\(3\sqrt{3}\)[/tex] and [tex]\(\sqrt{12}\)[/tex] with [tex]\(2\sqrt{3}\)[/tex].
- The expression becomes [tex]\(\sqrt{6} + 2\sqrt{3} + 3\sqrt{3} - 2\sqrt{3}\)[/tex].

4. Combine like terms:
- Combine the terms involving [tex]\(\sqrt{3}\)[/tex]:
[tex]\[
2\sqrt{3} + 3\sqrt{3} - 2\sqrt{3} = (2 + 3 - 2)\sqrt{3} = 3\sqrt{3}
\][/tex]

5. Final simplified expression:
- The expression simplifies to [tex]\(\sqrt{6} + 3\sqrt{3}\)[/tex].

Therefore, the expression [tex]\(\sqrt{6} + 2\sqrt{3} + \sqrt{27} - \sqrt{12}\)[/tex] simplifies to [tex]\(\sqrt{6} + 3\sqrt{3}\)[/tex].