Can we find $ a + b $?

Question

How can we find the value of $ a + b $ in the following question? a & b are integers.

Question: If $ a^{2} \times b^{3} = 216 $, find $ a + b $.

Answer 1

If the question is asking for non integer solutions:

$$b=(216/a^2)^{1/3}$$

Now add that to $a$.

If the question is asking for integer solutions we observe that:

$$\frac{216}{a^2}=b^3$$

Meaning that $a^2$ needs to be a factor of $216$, a perfect square, and when division is operated it must make a perfect cube.

We check the factors of $216$:

$$1,2,3,4,6,8,9,12,18,24,27,36,54,72,108,216$$

The only perfect squares are :

$$1,4,9,36$$

$$\frac{216}{1}=216=6^3$$

$$\frac{216}{4}=54$$

$$\frac{216}{9}=24$$

$$\frac{216}{36}=6$$

Thus the only thing that works from the above is:

$$a=1,b=6$$

$$a+b=7$$

But if you allow negative integer solutions:

$a=-1,b=6$ also works

$$a+b=5$$