Q: Is 5 5 11 larger than 2 2 2 4 2 ?

No, 2 2 2 4 2 is significantly larger. Despite the smaller base, the height and the values higher up in the stack of the second tower make it grow much faster than the first.

Question 1

What is a power tower?

Accepted Answer

A power tower is an exponential expression where the exponent itself is an exponential expression, and so on. For example, a^{b^c} means a raised to the power of (b raised to the power of c).

Question 2

How do you compare them without a calculator?

Accepted Answer

The general rule is that the taller the tower, the larger the number. For towers of the same height, the top-most difference usually decides the winner because exponents grow much faster than their bases. You can also use the rule that a^b is larger than c^d if b·log(a) is larger than d·log(c).

Question 3

Is 5^5¹¹ larger than 2^{2^{2^4²}}?

Accepted Answer

No, 2^{2^{2^4²}} is significantly larger. Despite the smaller base, the height and the values higher up in the stack of the second tower make it grow much faster than the first.

Question 4

Does it get harder?

Accepted Answer

Yes. Higher difficulty introduces taller towers and values that are closer together, requiring more careful logarithmic reasoning.

Question 5

Can I print it as a worksheet?

Accepted Answer

Yes. Power Towers supports the printable worksheet generator at /worksheets · each problem prints two towers with a lettered choice for the larger one and an answer key.