# Why is this not the simplest form of this expression?

The precalculus text states the solution to a geometric series problem as: $2\sqrt{6}(\sqrt{2}+1)$

Why wouldn't we carry out the multiplication and say that the answer in simplest form is:
$2\sqrt{6}(\sqrt{2}+1)=2\sqrt{12}+2\sqrt{6} = 4\sqrt{3}+2\sqrt{6}$

-
But why would the second form be simpler than the first? There are often advantages to factoring out common terms, as is done in the first answer. –  T. Bongers Jul 5 at 1:31
Hmmm...maybe I'm confused about what it means for an expression to be in simplest form. I guess I thought of addition as being simpler than multiplication. –  Nick Jul 5 at 1:35
There's no absolute rules about what is "simpler." There are good reasons to avoid sums of complicated terms, but a lot of it is just taste. –  T. Bongers Jul 5 at 1:37
Does the book actually say that $2\sqrt{6}(\sqrt{2}+1)$ is in simplest form? –  JimmyK4542 Jul 5 at 1:41
It does not, and so perhaps I was reading too much into it. –  Nick Jul 5 at 2:13

For the particular case of a geometric series with initial term $a$ and ratio $r$, remember that the sum of the first $n$ terms is $a\frac{1-r^n}{1-r}$. Depending on the particularities of the series, it could be that the textbook, by showing the final answer as a product, may make the connection between the first term of the series and the ratio $\frac{1-r^n}{1-r}$.

-
Interesting, I hadn't thought of that. The series to be solved was $\sqrt{12}+\sqrt{6}+\sqrt{3}+...$ Given that, I'm not seeing a connection between the first term and the ratio. Or am I overlooking it? –  Nick Jul 5 at 2:12
Here's what they did. It should be $\sqrt{12}(\sqrt{2}+2)$, where you can see $\sqrt{12}$ factoring out and the second term is indeed the the ratio (which for an infinite series is $1/(1-r)$). The ratio here is clearly $\sqrt{2}/2$. Now, they wrote $2$ as $\sqrt{2}\sqrt{2}$ and forced out another $\sqrt{2}$ to get your answer. I would not have done that; the result is more intuitive with $\sqrt{12}$ instead of $\sqrt{24}$. –  baudolino Jul 5 at 2:34

"Simplify" is a misleading term. What looks simpler to one person may be more complicated to someone else. Usually when an exercise in a book asks you to "simplify" it really means "do the operation we've been teaching you about in the chapter" such as factoring, expanding, etc. If it was a geometric series problem and they don't specifically tell you to simplify a certain way, either answer is fine.

-

The way I see it, as long as your answer is correct and does not have any terms that can be simplified (e.g. an answer with $3\sqrt 6+2\sqrt 6$ wouldn't be correct), it doesn't matter if it is factored or not. Personally, I like to see the factored form, but others may like to see the expanded form. If I was marking your work, and the problem did not specify to factor or expand, I would accept both answers.

-