To find the product and simplify the given expression [tex]\(\sqrt{y}(\sqrt{2y} + 2)\)[/tex], let's go through it step-by-step.
1. Distribute [tex]\(\sqrt{y}\)[/tex] to both terms inside the parentheses:
[tex]\[
\sqrt{y}(\sqrt{2y}) + \sqrt{y}(2)
\][/tex]
2. Simplify each term separately:
- The first term is [tex]\(\sqrt{y} \cdot \sqrt{2y}\)[/tex]:
[tex]\[
\sqrt{y} \cdot \sqrt{2y} = \sqrt{y \cdot 2y} = \sqrt{2y^2}
\][/tex]
- Since [tex]\(y\)[/tex] is nonnegative, [tex]\(\sqrt{y^2} = y\)[/tex]:
[tex]\[
\sqrt{2y^2} = \sqrt{2} \cdot \sqrt{y^2} = \sqrt{2} \cdot y = y\sqrt{2}
\][/tex]
- The second term is:
[tex]\[
\sqrt{y} \cdot 2 = 2\sqrt{y}
\][/tex]
3. Combine the simplified terms:
[tex]\[
\sqrt{y}(\sqrt{2y} + 2) = y\sqrt{2} + 2\sqrt{y}
\][/tex]
Therefore, the simplified expression is:
[tex]\[
y\sqrt{2} + 2\sqrt{y}
\][/tex]
This is the exact answer, using radicals as needed.
