Certainly! Let's tackle the expression step by step to simplify it.
The expression given is:
[tex]\[ \sqrt{2 a^3} - 3 \sqrt{2 a} + a \sqrt{8 a} + \sqrt{18 a} \][/tex]
Let's simplify each term individually:
1. First Term: [tex]\(\sqrt{2 a^3}\)[/tex]
[tex]\[
\sqrt{2 a^3} = \sqrt{2} \cdot \sqrt{a^3} = \sqrt{2} \cdot a^{3/2}
\][/tex]
2. Second Term: [tex]\( -3 \sqrt{2 a} \)[/tex]
[tex]\[
-3 \sqrt{2 a} = -3 \cdot \sqrt{2} \cdot \sqrt{a} = -3 \cdot \sqrt{2} \cdot a^{1/2}
\][/tex]
3. Third Term: [tex]\(a \sqrt{8 a}\)[/tex]
[tex]\[
a \sqrt{8 a} = a \cdot \sqrt{8} \cdot \sqrt{a} = a \cdot \sqrt{8} \cdot a^{1/2}
\][/tex]
Noting that [tex]\(\sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2}\)[/tex], we get:
[tex]\[
a \sqrt{8 a} = a \cdot 2\sqrt{2} \cdot a^{1/2} = 2a^{1 + 1/2} \cdot \sqrt{2} = 2 a^{3/2} \cdot \sqrt{2}
\][/tex]
4. Fourth Term: [tex]\(\sqrt{18 a}\)[/tex]
[tex]\[
\sqrt{18 a} = \sqrt{18} \cdot \sqrt{a} = \sqrt{9 \cdot 2} \cdot \sqrt{a} = 3\sqrt{2} \cdot \sqrt{a} = 3\sqrt{2} \cdot a^{1/2}
\][/tex]
Now, let's put all these simplified terms together:
[tex]\[
\sqrt{2} \cdot a^{3/2} - 3 \cdot \sqrt{2} \cdot a^{1/2} + 2 a^{3/2} \cdot \sqrt{2} + 3 \cdot \sqrt{2} \cdot a^{1/2}
\][/tex]
Combine like terms:
- The terms with [tex]\(a^{3/2}\)[/tex]:
[tex]\[
\sqrt{2} \cdot a^{3/2} + 2 \sqrt{2} \cdot a^{3/2} = (\sqrt{2} + 2\sqrt{2}) a^{3/2} = 3 \sqrt{2} \cdot a^{3/2}
\][/tex]
- The terms with [tex]\(a^{1/2}\)[/tex]:
[tex]\[
-3 \cdot \sqrt{2} \cdot a^{1/2} + 3 \cdot \sqrt{2} \cdot a^{1/2} = (-3 + 3) \sqrt{2} \cdot a^{1/2} = 0
\][/tex]
So, the simplified expression is:
[tex]\[
3 \sqrt{2} \cdot a^{3/2}
\][/tex]
Considering the structure, we notice that [tex]\(2 a^{3/2}\sqrt{2} + a^{3/2}\sqrt{2} = (2a^{3/2} + a^{3/2}) \sqrt{2} = \sqrt{2} (2a^{3/2} + a^{3/2})\)[/tex].
Thus, the final simplified form is:
[tex]\[
\sqrt{2}(2a^{3/2} + a^{3/2})
\][/tex]
