A Nice Exponential Equation. Solve x^x=2^2048.

Simple exponential equation. Solve for x.
घातीय समीकरणों को कैसे हल करें.
如何求解指数方程。
如何求解指數方程。
Cómo resolver ecuaciones exponenciales.

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奥林匹克数学技巧
奧林匹克數學技巧
truco de la olimpiada de matemáticas

[Physicsnerd]

For x^x=2^2048, let x = 2^a: also 2048=2^11
Then: (2^a)^(2^a)=2^(2^11)
And then: a*2^a=2^11
And then: a*2^a=2^3*2^8
And then: a*a^a=8*2^8
Therfore: a=8
And then: x=2^8

[Psykolord]

Before watching:

There are two similar ways of doing this that spring to mind.

We need to recall that a^(mn) = (a^m)^n. Further, 2048 is itself a power of 2, specifically 2^11.

Thus we can rewrite this as x^x = 2^(2^11) = 2^(2*2*2*2*2*2*2*2*2*2*2).

To get things in terms of base 4, we can rewrite the right as (2^2)^(2^10). We can do similar things, trying to reach a point of (2^n)^(2^n), as that will tell us our X.

(2^2)^(2^10) = (2^(2^2))^(2^9) = (2^(2^3))^(2^8)

Looking at this last portion, we know 2^3 = 8. Thus this can be written as (2^8)^(2^8), and thus x = 2^8. From here we simply evaluate the expression, and find that:

X = 256

[joseeduardomachado]

Só parece difícil, matemática é linda