To simplify the expression [tex]\((-9 \sqrt{2})(4 \sqrt{6})\)[/tex], follow these steps:
1. Multiply the constants: Multiply [tex]\(-9\)[/tex] by [tex]\(4\)[/tex]:
[tex]\[
-9 \times 4 = -36
\][/tex]
2. Multiply the square roots: Multiply [tex]\(\sqrt{2}\)[/tex] by [tex]\(\sqrt{6}\)[/tex]. Utilizing the property of square roots that [tex]\(\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}\)[/tex]:
[tex]\[
\sqrt{2} \times \sqrt{6} = \sqrt{2 \times 6} = \sqrt{12}
\][/tex]
3. Simplify [tex]\(\sqrt{12}\)[/tex]: Break down [tex]\(\sqrt{12}\)[/tex] into its prime factors, [tex]\(12 = 4 \times 3\)[/tex], so:
[tex]\[
\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2 \times \sqrt{3} = 2\sqrt{3}
\][/tex]
4. Combine results: The simplified form of [tex]\(\sqrt{12}\)[/tex] is [tex]\(2\sqrt{3}\)[/tex], now multiply this result by the previously obtained constant [tex]\(-36\)[/tex]:
[tex]\[
-36 \times 2\sqrt{3} = -72\sqrt{3}
\][/tex]
Therefore, the expression [tex]\((-9 \sqrt{2})(4 \sqrt{6})\)[/tex] simplified is:
[tex]\[
-72 \sqrt{3}
\][/tex]
So, the correct answer is:
[tex]\[
\text{B. } -72\sqrt{3}
\][/tex]
