Speed solving

Since x is a natural number , only one value of x satisfies the equation , that is 2 . Values more than 2 will result in the RHS of 3,4,5..... digits . $9^2 = 81$ $9 \times 2 = 18$

It is said that "ab" is a two-digit number. So,in 9^x, the only exponent that will make it a two-digit number is 2.

Since 9^x is a two digit integer, so :

x can't be 1 or 3, 4, ......

So x = 2 , which is valid also with the two digit integer 9 x

x = 2.

Ora, 9^2 = 81;

9 x 2 = 18.

A = 8;

B = 1.

9^2=81=AB, 9*2=18=BA

it is said that its a two digit number so 9^x must not 3 digit number so the solution is just simple putting x=2 we can find the answer.. ab=81 ,, ba=18

listing possible powers of 9 that is element of Z: 9 ^ 0 = 1 9 ^ 1 = 9 9 ^ 2 = 81 9 ^ 3 = 729 ... x would never be negative because 9^x cannot produce a negative value which can satisfy 9x notice that 9^2 is the only one that has 2 digits hence, x = 2, to check : 9^2 = 81 ; 9*2 = 18 where a = 8 and b = 1 in the form 10a+b

9^2=81
9x2=18

i'm good at prediction

Using trial and error method,first we go with the smallest no.

$9^{2} = 81$

And,

$9×2=18$

And also, you might get this instinctively.! ;)