<s>[INST] Struggling to simplify $w^{3/2}\sqrt{32} - w^{3/2}\sqrt{50}$ to $-w\sqrt{2w}$ I'm asked to simplify $w^{3/2}\sqrt{32} - w^{3/2}\sqrt{50}$ and am provided with the solution: $-w\sqrt{2w}$
I arrived at $9\sqrt{2}$ but I think I'm confused in understanding communitive rule here.
Here is my working:
$w^{3/2}\sqrt{32} - w^{3/2}\sqrt{50}$ = $\sqrt{w^3}\sqrt{32}$ - $\sqrt{w^3}\sqrt{50}$ # is this correct approach? I made the radical exponent a radical
Then:
$\sqrt{32}$ = $\sqrt{4}$ * $\sqrt{4}$ * $\sqrt{2}$ = $2 * 2 * \sqrt{2}$ = $4\sqrt{2}$
$\sqrt{50}$ = $\sqrt{2}$ * $\sqrt{25}$ = $5\sqrt{2}$
So:
$\sqrt{w^3}$$4\sqrt{2}$ - $\sqrt{w^3}5\sqrt{2}$ # should the expressions on either side of the minus sign be considered a single factor? i.e. could I also write as ($\sqrt{w^3}$$4\sqrt{2}$) - ($\sqrt{w^3}5\sqrt{2}$) )?
Then I'm less sure about where to go next. Since I have a positive $\sqrt{w^3}$ and a negative $\sqrt{w^3}$ I cancelled those out and was thus left with $9\sqrt{2}$.
More generally I was not sure of how to approach this and could not fin a justification for taking the path that I did.
How can I arrive at $-w\sqrt{2w}$ per the text book's solution?
 [/INST] Your approach is absolutely right. But note that $$
\begin{align}
\sqrt{w^3}4\sqrt{2}- \sqrt{w^3}5\sqrt{2}&=\sqrt{w^3}(4\sqrt{2}-5\sqrt{2})\\
&=-\sqrt{w^3}\sqrt{2}=-w\sqrt{2w}.
\end{align}$$
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