Andymath.com features free videos, notes, and practice problems with answers! Printable pages make math easy. Are you ready to be a mathmagician?

Practice Problems

\(\textbf{1)}\) \(2^{3^3}\)

Show Answer

The answer is \( 2^{27}=134{,}217{,}728 \)

Show Work

\(\,\,\,\,\,\,2^{3^3}=2^{\left(3^3\right)}\)

\(\,\,\,\,\,\,2^{3^3}=2^{27}\)

\(\,\,\,\,\,\,2^{3^3}=134{,}217{,}728\)

\(\textbf{2)}\) \(3^{2^4}\)

Show Answer

The answer is \( 3^{16}=43{,}046{,}721 \)

Show Work

\(\,\,\,\,\,\,3^{2^4}=3^{\left(2^4\right)}\)

\(\,\,\,\,\,\,3^{2^4}=3^{16}\)

\(\,\,\,\,\,\,3^{2^4}=43{,}046{,}721\)

\(\textbf{3)}\) \(2^{1^{1^{3}}}\)

Show Answer

The answer is \( 2 \)

Show Work

\(\,\,\,\,\,\,2^{1^{1^{3}}}=2^{\left(1^{1^{3}}\right)}\)

\(\,\,\,\,\,\,2^{1^{1^{3}}}=2^{\left(1^{1}\right)}\)

\(\,\,\,\,\,\,2^{1^{1^{3}}}=2^{1}\)

\(\,\,\,\,\,\,2^{1^{1^{3}}}=2\)

\(\textbf{4)}\) \(5^{2^{2}}\)

Show Answer

The answer is \( 5^{4}=625 \)

Show Work

\(\,\,\,\,\,\,5^{2^{2}}=5^{\left(2^{2}\right)}\)

\(\,\,\,\,\,\,5^{2^{2}}=5^{4}\)

\(\,\,\,\,\,\,5^{2^{2}}=625\)

\(\textbf{5)}\) \(4^{2^{3}}\)

Show Answer

The answer is \( 4^{8}=65{,}536 \)

Show Work

\(\,\,\,\,\,\,4^{2^{3}}=4^{\left(2^{3}\right)}\)

\(\,\,\,\,\,\,4^{2^{3}}=4^{8}\)

\(\,\,\,\,\,\,4^{2^{3}}=65{,}536\)

\(\textbf{6)}\) \(3^{3^{2}}\)

Show Answer

The answer is \( 3^{9}=19{,}683 \)

Show Work

\(\,\,\,\,\,\,3^{3^{2}}=3^{\left(3^{2}\right)}\)

\(\,\,\,\,\,\,3^{3^{2}}=3^{9}\)

\(\,\,\,\,\,\,3^{3^{2}}=19{,}683\)

\(\textbf{7)}\) \(9 = 3^{4^{\sqrt{2}^{x}}}\)

Show Answer

The answer is \(x=-2\)

Show Work

\(\,\,\,\,\,\,9 = 3^{4\sqrt{2}^{\,x}}\)

\(\,\,\,\,\,\,3^{2} = 3^{4\sqrt{2}^{\,x}}\)

\(\,\,\,\,\,\,2 = 4\sqrt{2}^{\,x}\)

\(\,\,\,\,\,\,4^{1/2} = 4^{\sqrt{2}^{\,x}}\)

\(\,\,\,\,\,\,\frac12 = \sqrt{2}^{\,x}\)

\(\,\,\,\,\,\,2^{-1} = 2^{x/2}\)

\(\,\,\,\,\,\,-1 = \frac{x}{2}\)

\(\,\,\,\,\,\,x=-2\)

\(\textbf{8)}\) \(\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}\,\cdots}}}\)

Show Answer

The answer is \( 2 \)

Show Work

\(\,\,\,\,\,\,\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}\,\cdots}}} = x\)

\(\,\,\,\,\,\,x = \left(\sqrt{2}\right)^{\sqrt{2}^{\sqrt{2}^{\cdots}}}\)

\(\,\,\,\,\,\,x = (\sqrt{2})^{x}\)

\(\,\,\,\,\,\,x = (2^{1/2})^{x}\)

\(\,\,\,\,\,\,x = 2^{x/2}\)

\(\,\,\,\,\,\,2 = 2^{x/2}\)

\(\,\,\,\,\,\,2^{1} = 2^{x/2}\)

\(\,\,\,\,\,\,1 = \frac{x}{2}\)

\(\,\,\,\,\,\,x = 2\)

See Related Pages\(\)

\(\bullet\text{ Intro to Exponents}\)
\(\,\,\,\,\,\,\,\,3^2=3 \times 3 =9\)

\(\bullet\text{ Negative Exponents}\)
\(\,\,\,\,\,\,\,\,3^{-2}=\frac{1}{9}\)

\(\bullet\text{ Rational Exponents}\)
\(\,\,\,\,\,\,\,\,\displaystyle x^{a/b}=\sqrt[b]{x^a}\)

In Summary

Power towers—also known as iterated exponentials—are expressions where exponents stack on top of each other, such as \(\,a^{b^{c^{d}}}.\,\) Unlike regular exponentiation, power towers must be evaluated from the top down, because each exponent itself may contain another exponent. This top-down structure makes them grow extremely quickly and gives them interesting mathematical properties.

Power towers are introduced informally in advanced algebra or precalculus settings, and they often appear in contest math, discrete mathematics, and early introductions to tetration. They help build intuition about exponential growth, order of operations, and how small changes in an exponent can lead to massive differences in final value.

The most common mistakes with power towers involve evaluating from the bottom up instead of from the top down, misunderstanding parentheses, or incorrectly simplifying intermediate exponents. With practice, students learn to identify the highest exponent first and work downward.

Mathematicians including Leonhard Euler and Jonathan Binet studied early forms of power towers while investigating exponential functions and infinite exponentials. Today, power towers and tetration remain a fascinating topic in recreational mathematics, number theory, and complexity studies due to their rapid growth and elegant structure.

About Andymath.com

Andymath.com is a free math website with the mission of helping students, teachers and tutors find helpful notes, useful sample problems with answers including step by step solutions, and other related materials to supplement classroom learning. If you have any requests for additional content, please contact Andy at tutoring@andymath.com. He will promptly add the content.

Topics cover Elementary Math, Middle School, Algebra, Geometry, Algebra 2/Pre-calculus/Trig, Calculus and Probability/Statistics. In the future, I hope to add Physics and Linear Algebra content.

Visit me on Youtube, Tiktok, Instagram and Facebook. Andymath content has a unique approach to presenting mathematics. The clear explanations, strong visuals mixed with dry humor regularly get millions of views. We are open to collaborations of all types, please contact Andy at tutoring@andymath.com for all enquiries. To offer financial support, visit my Patreon page. Let’s help students understand the math way of thinking!

Thank you for visiting. How exciting!