Let's simplify the expression [tex]\(\sqrt{24a^2}\)[/tex] step by step:
1. Simplify the expression inside the square root:
The given expression is [tex]\( \sqrt{24a^2} \)[/tex]. We know that 24 can be factored into 4 and 6:
[tex]\[ 24 = 4 \times 6 \][/tex]
Thus, we rewrite the expression inside the square root:
[tex]\[ 24a^2 = 4 \cdot 6 \cdot a^2 \][/tex]
2. Apply the square root to each factor:
Next, we apply the property of square roots which allows us to take the square root of each factor separately:
[tex]\[ \sqrt{24a^2} = \sqrt{4 \cdot 6 \cdot a^2} \][/tex]
Using the property [tex]\(\sqrt{xy} = \sqrt{x} \cdot \sqrt{y}\)[/tex], we get:
[tex]\[ \sqrt{4 \cdot 6 \cdot a^2} = \sqrt{4} \cdot \sqrt{6} \cdot \sqrt{a^2} \][/tex]
3. Simplify each square root:
We know:
[tex]\[ \sqrt{4} = 2 \][/tex]
[tex]\[ \sqrt{a^2} = a \][/tex]
So, we now have:
[tex]\[ \sqrt{4} \cdot \sqrt{6} \cdot \sqrt{a^2} = 2 \cdot \sqrt{6} \cdot a \][/tex]
4. Write the simplified expression:
Combining the results, we get:
[tex]\[ \sqrt{24a^2} = 2a \sqrt{6} \][/tex]
Now, let us compare this simplified expression with the given choices:
- [tex]\(A. \)[/tex] [tex]\(2 a^3 \sqrt{6 a}\)[/tex]
- [tex]\(B. \)[/tex] [tex]\(6 a \sqrt{2 a^3}\)[/tex]
- [tex]\(C. \)[/tex] [tex]\(4 a^3 \sqrt{6 a}\)[/tex]
- [tex]\(D. \)[/tex] [tex]\(4 a \sqrt{6 a^3}\)[/tex]
None of these choices directly match the simplified form [tex]\(2a \sqrt{6}\)[/tex].
Therefore, the correct choice is not available among the given options. This indicates an issue with the provided choices, suggesting that none of them correctly represent the simplified expression [tex]\(\sqrt{24a^2}\)[/tex]. Hence, the correct choice is not listed.
