Let's solve the given expression step-by-step:
We have the expression:
[tex]\[ 3 \sqrt{2 y^3} \cdot 7 \sqrt{18 y} \][/tex]
First, simplify each square root term separately.
1. Simplify [tex]\( \sqrt{2 y^3} \)[/tex]:
[tex]\[
\sqrt{2 y^3} = \sqrt{2} \cdot \sqrt{y^3}
\][/tex]
We know that [tex]\( \sqrt{y^3} = y^{1.5} \)[/tex] because [tex]\( \sqrt{y^3} = y^{3/2} \)[/tex]:
[tex]\[
\sqrt{2 y^3} = \sqrt{2} \cdot y^{3/2}
\][/tex]
2. Simplify [tex]\( \sqrt{18 y} \)[/tex]:
[tex]\[
\sqrt{18 y} = \sqrt{18} \cdot \sqrt{y}
\][/tex]
We know that [tex]\( 18 \)[/tex] can be factored into [tex]\( 2 \cdot 9 \)[/tex], and [tex]\( \sqrt{18} = \sqrt{2 \cdot 9} = \sqrt{2} \cdot \sqrt{9} = \sqrt{2} \cdot 3 \)[/tex]:
[tex]\[
\sqrt{18 y} = 3 \sqrt{2} \cdot \sqrt{y} = 3 \sqrt{2} \cdot y^{1/2}
\][/tex]
Now, multiply the simplified square roots by their coefficients:
[tex]\[
3 \sqrt{2} \cdot y^{3/2} \quad \text{and} \quad 7 \cdot 3 \sqrt{2} \cdot y^{1/2}
\][/tex]
So, we have:
[tex]\[
3 \sqrt{2} \cdot y^{3/2} \cdot 7 \cdot 3 \sqrt{2} \cdot y^{1/2}
\][/tex]
Combine the constants together:
[tex]\[
3 \cdot 7 \cdot 3 = 63
\][/tex]
And the square root terms:
[tex]\[
\sqrt{2} \cdot \sqrt{2} = 2
\][/tex]
Multiply these:
[tex]\[
63 \cdot 2 = 126
\][/tex]
Combine the [tex]\( y \)[/tex]-terms:
[tex]\[
y^{3/2} \cdot y^{1/2} = y^{(3/2 + 1/2)} = y^2
\][/tex]
Putting all parts together:
[tex]\[
126 \cdot y^2
\][/tex]
Therefore, the expression simplifies to:
[tex]\[
126 y^2
\][/tex]
The correct answer is:
D. [tex]\( 126 y^2 \)[/tex]
