$\large a^b = 1024$

How many integers $(a,b)$ exist for this equation?

**
Note:
**
$b \ne 1$

The answer is 5.

**
This section requires Javascript.
**

You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.

We know that $1024 = 2^ {10} = 2^{2\times 5}$ , hence either $a=2$ and $b=10$ or $a=2^2 =4$ and $b=5$ or $a=2^5=32$ and $b=2$ .

Don't forget about negative integers : $(-2)^{10}= 2^{10}$ and $(-32)^2 = 32^2$ .

So in total $5$ couples of integers $(a,b)$ satisfy the equation : $\{(-32,2),(-2,10),(2,10),(4,5),(32,2)\}$ .