7a⁴√2a

To simplify the expression \( 7a^4\sqrt{2a} \), we will separate the components.

1. Rewrite \(\sqrt{2a}\) as \(\sqrt{2} \cdot \sqrt{a}\).

2. So, the expression becomes: \[ 7a^4\sqrt{2} \cdot \sqrt{a} \]

3. Next, we can combine \(a^4\) and \(\sqrt{a}\):

   • Since \(\sqrt{a} = a^{1/2}\), we get: \[ a^4 \cdot a^{1/2} = a^{4 + 1/2} = a^{4.5} = a^{\frac{9}{2}} \]

4. Therefore, the expression simplifies to: \[ 7\sqrt{2}a^{\frac{9}{2}} \]

So the simplified form of \( 7a^4\sqrt{2a} \) is \( 7\sqrt{2}a^{\frac{9}{2}} \).