$$x^{x^{x^{x^{\cdots x^{x^3}}}}}=3$$
Hint
No matter how many x's are there, the answer would be $$\sqrt[3]{3}$$
What?Thanic Nur Samin wrote:If x is even, then after 3 steps it would go imaginary.
What?Thanic Nur Samin wrote:If x is odd, then $x^{x^{x^{x^{\cdots x^{x^{3}}}}}}$ would be negative.
What?Thanic Nur Samin wrote:If x is a fraction, then $x^{x^{x^{x^{\cdots x^{x^{3}}}}}}$ would be negative.
Proof?Thanic Nur Samin wrote:If $x>1$, then,
$x^{x^{x^{x^{\cdots x^{x^{3}}}}}}$ is a stictly increasing function
Note what is written at the top of all those 3 cases.*Mahi* wrote:What?Thanic Nur Samin wrote:If x is even, then after 3 steps it would go imaginary.What?Thanic Nur Samin wrote:If x is odd, then $x^{x^{x^{x^{\cdots x^{x^{3}}}}}}$ would be negative.What?Thanic Nur Samin wrote:If x is a fraction, then $x^{x^{x^{x^{\cdots x^{x^{3}}}}}}$ would be negative.
For the first three cases, are you missing some log somewhere? Because $\text{positive}^{\text{anything}} > 0$ and is real.