How many layers are there?

Algebra Level 3

It's obvious that $4=2^2$ .

Can we also express $4^{4^4}$ as $\Large \underbrace{2^{2^{2^{\cdot^{\cdot^2}}}}}_{n \text{ 2's}}$ ?

No Yes

1 solution

Brian Charlesworth
Apr 28, 2017

$\Large 4^{4^{4}} = 4^{(2^{2})^{4}} = 4^{2^{8}} = (2^{2})^{2^{8}} = 2^{2^{9}}$ .

Since $9$ cannot be expressed as a "power tower" of $2$ 's, $4^{4^{4}}$ cannot be expressed as a power tower of $2$ 's either, so the answer is $\boxed{\text{No}}$ .

@Pi Han Goh The answer if fine, but I just thought I should point out that in your second example, $\Large4^{4} = 2^{8} = 2^{2^{3}}$ and not $\Large 2^{2^{2^{2}}} = 2^{2^{4}}$ .

Brian Charlesworth - 4 years, 1 month ago

It actually made myself doubt myself... But i won my internal debate ;)

Peter van der Linden - 4 years, 1 month ago

Sorry guys, I don't know indices.

Feel free to submit a report.

Pi Han Goh - 4 years, 1 month ago

Thanks. I've removed the second paragraph.

In future, if you spot any errors with a problem, you can “report” it by selecting "report problem" in the “line line line” menu in the bottom right corner. This will notify the problem creator who can fix the issues.

Brilliant Mathematics Staff - 4 years, 1 month ago

@Brilliant Mathematics – Yeah I know but I read Brian's comment and thought them it won't be needed

Peter van der Linden - 4 years, 1 month ago

How many layers are there?

1 solution

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