Answer :

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].