To simplify the expression [tex]\(\sqrt{24} - 2 \sqrt{12} + 7 \sqrt{6}\)[/tex], we need to break down each term into its simplest form.
### Step 1: Simplify [tex]\(\sqrt{24}\)[/tex]
First, we recognize that 24 can be factored as [tex]\(4 \times 6\)[/tex]. This allows us to use the property of square roots:
[tex]\[
\sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6} = 2 \sqrt{6}
\][/tex]
### Step 2: Simplify [tex]\(2 \sqrt{12}\)[/tex]
Next, we factor 12 as [tex]\(4 \times 3\)[/tex]:
[tex]\[
\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2 \sqrt{3}
\][/tex]
Thus,
[tex]\[
2 \sqrt{12} = 2 \times 2 \sqrt{3} = 4 \sqrt{3}
\][/tex]
### Step 3: Simplify [tex]\(7 \sqrt{6}\)[/tex]
The term [tex]\(7 \sqrt{6}\)[/tex] is already simplified.
### Step 4: Combine all simplified terms
Now we substitute the simplified terms back into the original expression:
[tex]\[
\sqrt{24} - 2 \sqrt{12} + 7 \sqrt{6} = 2 \sqrt{6} - 4 \sqrt{3} + 7 \sqrt{6}
\][/tex]
Next, we combine like terms. The like terms in this expression are the ones involving [tex]\(\sqrt{6}\)[/tex]:
[tex]\[
2 \sqrt{6} + 7 \sqrt{6} = 9 \sqrt{6}
\][/tex]
So, combining everything together, we get:
[tex]\[
9 \sqrt{6} - 4 \sqrt{3}
\][/tex]
### Conclusion
The simplified form of the expression [tex]\(\sqrt{24} - 2 \sqrt{12} + 7 \sqrt{6}\)[/tex] is:
[tex]\[
9 \sqrt{6} - 4 \sqrt{3}
\][/tex]
Therefore, the correct answer is:
A. [tex]\(9 \sqrt{6} - 4 \sqrt{3}\)[/tex]
