Operations with surds 
If $a=3\sqrt2,$ simplify $(2a)^3$

(Original screenshots: 1 2)
I have tried this even on my calculator and got it wrong, please show me the correct methodology. 
I  did $(2)^3 + (3\sqrt2)^3$ and got wrong answer
Then: $(3\sqrt2)^3 \cdot (2)^3$
and got wrong asnwer too, what am I doing wrong?
 A: You have made two errors.  The main one is that the formula $2a$ does not mean $2+a$.  It means 2 multiplied by $a$, that is $2\cdot a$.  To compute this on a calculator, you need to take $a$, which is $3\sqrt2$—that is, $3$ multiplied by $\sqrt 2$—and multiply that by $2$.  The result, $2a$, will be around 8.4.
Then you cube $2a$, which means you calculate $(2a)\times(2a)\times(2a)$.  The result should be around 600 or so.

To compute this symbolically, you start by substituting $a$ with its definition: $(2a)^3$ becomes $$(2\cdot 3\cdot\sqrt2)^3.$$
We can combine the 2 and 3 right away, because we know $2\cdot 3 = 6$: $$(6\cdot\sqrt 2)^3.$$
There is a rule that says that $(p\cdot q)^r = p^r\cdot q^r$. Here we have $p=6, q=\sqrt 2,$ and $r=3$:
$$6^3 \cdot (\sqrt 2)^3$$
Then $6^3 = 216$ and $(\sqrt2)^3 = \sqrt2\cdot\sqrt2\cdot\sqrt2 = 2\cdot \sqrt 2$, so we get:
$$216\cdot 2\cdot\sqrt 2 = 432\cdot \sqrt 2$$
This is as simple as we can make the answer.

The other error you made is not important, in light of the first one, but it may trip you up later, so I mention it anyway.  If you did need to compute $(2 + a)^3$, you cannot compute the right answer by computing $2^3$ and $a^3$ separately and adding the results.  $(2+a)^3$ means that you do the addition before the cube, not the other way around, and in this case the order makes a difference, as your calculator will confirm.
It is easy to confuse this with the situation where you need to compute $(2\cdot a)^3$. There, it does work to cube $2$ and $a$ separately and multiply the results; you get the same answer as if you had multiplied $2$ and $a$ first and cubed the result.
