Certainly! Let's simplify the expression [tex]\(3 \sqrt{2 y^3} \cdot 7 \sqrt{18 y}\)[/tex].
1. Simplify each square root separately.
- First, simplify [tex]\(\sqrt{2 y^3}\)[/tex]:
[tex]\[
\sqrt{2 y^3} = \sqrt{2 y^2 \cdot y} = \sqrt{2 y^2} \cdot \sqrt{y} = y \sqrt{2} \cdot \sqrt{y} = y \sqrt{2 y}
\][/tex]
- Next, simplify [tex]\(\sqrt{18 y}\)[/tex]:
[tex]\[
\sqrt{18 y} = \sqrt{9 \cdot 2 y} = \sqrt{9} \cdot \sqrt{2 y} = 3 \sqrt{2 y}
\][/tex]
2. Substitute the simplified forms back into the expression:
[tex]\[
3 \sqrt{2 y^3} \cdot 7 \sqrt{18 y} = 3 (y \sqrt{2 y}) \cdot 7 (3 \sqrt{2 y})
\][/tex]
3. Combine the constants and the terms involving [tex]\(y\)[/tex] and [tex]\(\sqrt{2 y}\)[/tex]:
- Constants: [tex]\(3 \cdot 7 \cdot 3 = 63\)[/tex]
- Terms involving [tex]\(y\)[/tex] and [tex]\(\sqrt{2 y}\)[/tex]:
[tex]\[
\sqrt{2 y} \cdot \sqrt{2 y} = \sqrt{4 y^2} = 2 y
\][/tex]
4. Put it all together:
[tex]\[
63 \cdot y \cdot 2 y = 63 \cdot 2 y^2 = 126 y^2
\][/tex]
Thus, the equivalent expression to [tex]\(3 \sqrt{2 y^3} \cdot 7 \sqrt{18 y}\)[/tex] is [tex]\(126 y^2\)[/tex].
The correct answer is \(D. 126 y^2)\
