Sure, let's simplify the given expression step-by-step.
Given: [tex]\(\sqrt{2a}\left(2 \sqrt{8a} + \sqrt{2a^3}\right)\)[/tex]
1. Simplify [tex]\(\sqrt{2a} \cdot 2\sqrt{8a}\)[/tex]
- First, consider [tex]\(\sqrt{8a}\)[/tex]:
[tex]\[
\sqrt{8a} = \sqrt{4 \cdot 2a} = \sqrt{4} \cdot \sqrt{2a} = 2\sqrt{2a}
\][/tex]
- Next, multiply by 2:
[tex]\[
2\sqrt{8a} = 2 \cdot 2\sqrt{2a} = 4\sqrt{2a}
\][/tex]
2. Now, simplify [tex]\(\sqrt{2a} \cdot \sqrt{2a^3}\)[/tex]
- Consider [tex]\(\sqrt{2a^3}\)[/tex]:
[tex]\[
\sqrt{2a^3} = \sqrt{2a^2 \cdot a} = \sqrt{2a^2} \cdot \sqrt{a} = a\sqrt{2a}
\][/tex]
- Now, multiply:
[tex]\[
\sqrt{2a} \cdot \sqrt{2a^3} = \sqrt{2a} \cdot (a\sqrt{2a}) = a (\sqrt{2a} \cdot \sqrt{2a}) = a \cdot \sqrt{4a^2} = a \cdot 2a = 2a^2
\][/tex]
3. Combine the parts:
Now we have:
[tex]\[
\sqrt{2a} \left(4\sqrt{2a} + a\sqrt{2a}\right) = \sqrt{2a}(4\sqrt{2a}) + \sqrt{2a}(a\sqrt{2a})
\][/tex]
Simplify each term:
[tex]\[
4\sqrt{2a} \cdot \sqrt{2a} = 4 \cdot (\sqrt{2a})^2 = 4 \cdot 2a = 8a
\][/tex]
[tex]\[
\sqrt{2a} \cdot (a\sqrt{2a}) = a \cdot \sqrt{2a} \cdot \sqrt{2a} = a \cdot 2a = 2a^2
\][/tex]
Adding these terms together:
[tex]\[
8a + 2a^2
\][/tex]
Hence, the simplified expression is:
[tex]\[
\boxed{2a^2 + 8|a|}
\][/tex]
