Practice Problems

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

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

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\(\,\,\,\,\,\,2^{3^3}=2^{\left(3^3\right)}\)
\(\,\,\,\,\,\,2^{3^3}=2^{27}\)
\(\,\,\,\,\,\,2^{3^3}=134{,}217{,}728\)

 

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

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

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\(\,\,\,\,\,\,3^{2^4}=3^{\left(2^4\right)}\)
\(\,\,\,\,\,\,3^{2^4}=3^{16}\)
\(\,\,\,\,\,\,3^{2^4}=43{,}046{,}721\)

 

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

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The answer is \( 2 \)

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

 

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

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

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\(\,\,\,\,\,\,5^{2^{2}}=5^{\left(2^{2}\right)}\)
\(\,\,\,\,\,\,5^{2^{2}}=5^{4}\)
\(\,\,\,\,\,\,5^{2^{2}}=625\)

 

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

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

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\(\,\,\,\,\,\,4^{2^{3}}=4^{\left(2^{3}\right)}\)
\(\,\,\,\,\,\,4^{2^{3}}=4^{8}\)
\(\,\,\,\,\,\,4^{2^{3}}=65{,}536\)

 

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

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

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\(\,\,\,\,\,\,3^{3^{2}}=3^{\left(3^{2}\right)}\)
\(\,\,\,\,\,\,3^{3^{2}}=3^{9}\)
\(\,\,\,\,\,\,3^{3^{2}}=19{,}683\)

 

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

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The answer is \(x=-2\)

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\(\,\,\,\,\,\,9 = 3^{4\sqrt{2}^{\,x}}\)
\(\,\,\,\,\,\,3^{2} = 3^{4\sqrt{2}^{\,x}}\)
\(\,\,\,\,\,\,2 = 4\sqrt{2}^{\,x}\)
\(\,\,\,\,\,\,4^{1/2} = 4^{\sqrt{2}^{\,x}}\)
\(\,\,\,\,\,\,\frac12 = \sqrt{2}^{\,x}\)
\(\,\,\,\,\,\,2^{-1} = 2^{x/2}\)
\(\,\,\,\,\,\,-1 = \frac{x}{2}\)
\(\,\,\,\,\,\,x=-2\)

 

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

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The answer is \( 2 \)

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\(\,\,\,\,\,\,\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}\,\cdots}}} = x\)
\(\,\,\,\,\,\,x = \left(\sqrt{2}\right)^{\sqrt{2}^{\sqrt{2}^{\cdots}}}\)
\(\,\,\,\,\,\,x = (\sqrt{2})^{x}\)
\(\,\,\,\,\,\,x = (2^{1/2})^{x}\)
\(\,\,\,\,\,\,x = 2^{x/2}\)
\(\,\,\,\,\,\,2 = 2^{x/2}\)
\(\,\,\,\,\,\,2^{1} = 2^{x/2}\)
\(\,\,\,\,\,\,1 = \frac{x}{2}\)
\(\,\,\,\,\,\,x = 2\)

 

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.