To find which choice is equivalent to the given expression for [tex]\( y \geq 0 \)[/tex], we need to simplify the expression:
[tex]\[ \sqrt{y^3} + \sqrt{4y^3} - 2y \sqrt{y} \][/tex]
First, let's break down each term:
1. [tex]\(\sqrt{y^3}\)[/tex]:
- The expression [tex]\(y^3\)[/tex] inside the square root can be rewritten as [tex]\((y^2) \cdot y\)[/tex].
- We know that [tex]\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)[/tex]. Applying this, we get:
[tex]\[
\sqrt{y^3} = \sqrt{(y^2) \cdot y} = \sqrt{y^2} \cdot \sqrt{y}
\][/tex]
- Since [tex]\(\sqrt{y^2} = y\)[/tex] for [tex]\(y \geq 0\)[/tex], it follows that:
[tex]\[
\sqrt{y^3} = y \sqrt{y}
\][/tex]
2. [tex]\(\sqrt{4y^3}\)[/tex]:
- Similarly, [tex]\(4y^3\)[/tex] can be rewritten as [tex]\(4 \cdot y^3 = 4 \cdot (y^2) \cdot y\)[/tex].
- Applying the property of square roots again:
[tex]\[
\sqrt{4y^3} = \sqrt{4 \cdot (y^2 \cdot y)} = \sqrt{4} \cdot \sqrt{y^2 \cdot y}
\][/tex]
- Since [tex]\(\sqrt{4} = 2\)[/tex] and [tex]\(\sqrt{y^2} = y\)[/tex], we get:
[tex]\[
\sqrt{4y^3} = 2 \cdot y \sqrt{y} = 2y \sqrt{y}
\][/tex]
3. [tex]\(-2y \sqrt{y}\)[/tex]:
- This term is already simplified.
Putting it all together, the expression becomes:
[tex]\[
\sqrt{y^3} + \sqrt{4y^3} - 2y \sqrt{y} = y \sqrt{y} + 2y \sqrt{y} - 2y \sqrt{y}
\][/tex]
Notice that [tex]\(2y \sqrt{y}\)[/tex] and [tex]\(-2y \sqrt{y}\)[/tex] cancel each other out, leaving us with:
[tex]\[
y \sqrt{y}
\][/tex]
Thus, the simplified form of the given expression is:
[tex]\[
y \sqrt{y}
\][/tex]
Therefore, the correct choice equivalent to the expression is:
[tex]\[
\boxed{D} \quad y \sqrt{y}
\][/tex]
