To simplify the expression [tex]\(\sqrt{12} \cdot 4 \sqrt{3}\)[/tex], follow these steps:
1. Simplify [tex]\(\sqrt{12}\)[/tex]:
Break down the number inside the square root into its prime factors.
[tex]\[ \sqrt{12} = \sqrt{4 \cdot 3} \][/tex]
Since the square root of a product is the product of the square roots,
[tex]\[ \sqrt{12} = \sqrt{4} \cdot \sqrt{3} \][/tex]
And knowing that [tex]\(\sqrt{4} = 2\)[/tex],
[tex]\[ \sqrt{12} = 2 \sqrt{3} \][/tex]
2. Substitute the simplified [tex]\(\sqrt{12}\)[/tex] back into the expression:
[tex]\[ 2 \sqrt{3} \cdot 4 \sqrt{3} \][/tex]
3. Combine the terms:
Here, we can multiply the constants together and the square root terms together separately.
[tex]\[ (2 \cdot 4) \cdot (\sqrt{3} \cdot \sqrt{3}) \][/tex]
The constants multiply to give:
[tex]\[ 2 \cdot 4 = 8 \][/tex]
And [tex]\(\sqrt{3} \cdot \sqrt{3}\)[/tex] simplifies to:
[tex]\[ \sqrt{3} \cdot \sqrt{3} = 3 \][/tex]
4. Multiply the results from the constants and square roots:
[tex]\[ 8 \cdot 3 = 24 \][/tex]
Therefore, the simplified form of the expression [tex]\(\sqrt{12} \cdot 4 \sqrt{3}\)[/tex] is:
[tex]\[ \boxed{24} \][/tex]
