Solve x^z = 4096?

Question

x and z both are integers

we were told there were 12 solutions...?

Answer 1

There are numerous solutions:
x = 2,z = 12
x = 4,z = 6
x = 8,z = 4
x = 16,z = 3
x = 64,z = 2
x = 4096,z = 1

All these stem from that fact that 4096 = 2^12 and you can factorise 12 in several different ways.
I.e.
2^12 = 2^(1*12) = (2^1)^12 = 2^12
2^12 = 2^(2*6) = (2^2)^6 = 4^6
2^12 = 2^(3*4) = (2^3)^4 = 8^4
2^12 = 2^(4*3) = (2^4)^3 = 16^3
2^12 = 2^(6*2) = (2^6)^2 = 64^2
2^12 = 2^(12*1) = (2^12)^1 = 4096^1

Additionally you could also have:
x = -2, z = 12
x = -4, z = 6
x = -8, z = 4
x = -64, z = 2
For a grand total of 10 unique solutions.

Answer 2

64^2 = 4096
16^3 = 4096
8^4 = 4096
4^6 = 4096
2^12 = 4096

Answer 3

64^2 = 4096

Answer 4

Possible answers:
64 ^2
16^3
8^4
4^6

as x and z are integers the answer is either 8⁴ or 4^6

ANS: x = 8 & z=4 OR x=4 & z=6

Answer 5

x=4096^(1/z)
z=[ln(2)]/ln(x)

you could put any number in for one and solve for the other.

Answer 6

You have 2 variables and 1 equation. A solution is not possible.
z ln (x) = ln (4096)
ln (x) = ln (4096)/z
x = exp(ln (4096)/z), z unknown.

Similarly, if you solve for z, x is the unknown.

Answer 7

64^2 = 4096

x = 64
z = 2