Let's solve the given expression step-by-step:
Given expression:
[tex]\[ \sqrt{2a^3} - 3 \sqrt{2a} + a \sqrt{8a} + \sqrt{18a} \][/tex]
Our goal is to simplify this expression.
### Step 1: Simplify each term individually
1. Simplify [tex]\(\sqrt{2a^3}\)[/tex]:
[tex]\[
\sqrt{2a^3} = \sqrt{2} \cdot \sqrt{a^3} = \sqrt{2} \cdot a^{3/2}
\][/tex]
2. Simplify [tex]\(\sqrt{2a}\)[/tex]:
[tex]\[
\sqrt{2a} = \sqrt{2} \cdot \sqrt{a} = \sqrt{2} \cdot a^{1/2}
\][/tex]
3. Simplify [tex]\(a \sqrt{8a}\)[/tex]:
[tex]\[
a \sqrt{8a} = a \cdot \sqrt{8} \cdot \sqrt{a} = a \cdot \sqrt{8} \cdot a^{1/2} = a \cdot \sqrt{4 \cdot 2} \cdot a^{1/2} = a \cdot 2\sqrt{2} \cdot a^{1/2} = 2 \cdot \sqrt{2} \cdot a \cdot a^{1/2} = 2 \sqrt{2} \cdot a^{3/2}
\][/tex]
4. Simplify [tex]\(\sqrt{18a}\)[/tex]:
[tex]\[
\sqrt{18a} = \sqrt{18} \cdot \sqrt{a} = \sqrt{9 \cdot 2} \cdot a^{1/2} = 3 \sqrt{2} \cdot a^{1/2}
\][/tex]
### Step 2: Substitute these simplifications back into the expression
Now we substitute back each simplified term into the original expression:
[tex]\[
\sqrt{2} \cdot a^{3/2} - 3 \sqrt{2} \cdot a^{1/2} + 2 \sqrt{2} \cdot a^{3/2} + 3 \sqrt{2} \cdot a^{1/2}
\][/tex]
### Step 3: Combine like terms
Combine the terms with [tex]\(\sqrt{2} \cdot a^{3/2}\)[/tex] and with [tex]\(\sqrt{2} \cdot a^{1/2}\)[/tex]:
[tex]\[
(\sqrt{2} \cdot a^{3/2} + 2 \sqrt{2} \cdot a^{3/2}) + (-3 \sqrt{2} \cdot a^{1/2} + 3 \sqrt{2} \cdot a^{1/2})
\][/tex]
[tex]\[
= (1 \sqrt{2} \cdot a^{3/2} + 2 \sqrt{2} \cdot a^{3/2}) + (0 \cdot \sqrt{2} \cdot a^{1/2})
\][/tex]
Now combine the coefficients:
[tex]\[
= 3 \sqrt{2} \cdot a^{3/2}
\][/tex]
Therefore, the simplified form of the given expression is:
[tex]\[
\boxed{\sqrt{2} (2a^{3/2} + \sqrt{a^3})}
\][/tex]
In conclusion, the simplified expression for [tex]\(\sqrt{2a^3} - 3 \sqrt{2a} + a \sqrt{8a} + \sqrt{18a}\)[/tex] is [tex]\(\sqrt{2} (2a^{3/2} + \sqrt{a^3})\)[/tex].
