To solve the given problem [tex]\( 4 \sqrt{2} \times 6 \sqrt{18} \)[/tex], follow these steps:
1. Separate Constants and Radicals:
- The given expression is [tex]\( 4 \sqrt{2} \times 6 \sqrt{18} \)[/tex].
- Rewrite this as [tex]\( (4 \times 6) \times (\sqrt{2} \times \sqrt{18}) \)[/tex].
2. Multiply the Constants:
- The constants [tex]\(4\)[/tex] and [tex]\(6\)[/tex] multiply to give [tex]\(4 \times 6 = 24\)[/tex].
3. Multiply the Square Roots:
- The radicals [tex]\( \sqrt{2} \times \sqrt{18} \)[/tex] can be simplified by using the property of square roots: [tex]\( \sqrt{a} \times \sqrt{b} = \sqrt{a \times b} \)[/tex].
- Therefore, [tex]\( \sqrt{2} \times \sqrt{18} = \sqrt{2 \times 18} = \sqrt{36} \)[/tex].
4. Simplify the Square Root:
- Next, simplify [tex]\( \sqrt{36} \)[/tex]. Since [tex]\(36\)[/tex] is a perfect square, [tex]\( \sqrt{36} = 6 \)[/tex].
5. Final Multiplication:
- Now, multiply the constant [tex]\( 24 \)[/tex] by the simplified square root [tex]\( 6 \)[/tex]:
[tex]\[
24 \times 6 = 144
\][/tex]
Thus, the complete simplified expression of [tex]\( 4 \sqrt{2} \times 6 \sqrt{18} \)[/tex] is [tex]\( 144 \)[/tex].
Therefore, the correct answer is:
[tex]\[
\boxed{144}
\][/tex]
