Power towers are expressions where exponents are stacked, such as \(a^{b^c}\), and they are evaluated from the top down. Parentheses and the placement of exponents matter because changing the order can dramatically change the value of an expression. These practice problems include evaluating finite power towers, solving equations involving stacked exponents, and exploring infinite power towers.
Practice Problems
\(\textbf{1)}\) \(2^{3^3}\)The answer is \(2^{27}=134{,}217{,}728\)
\(\,\,\,\,\,\,2^{3^3}=2^{\left(3^3\right)}\)
\(\,\,\,\,\,\,2^{3^3}=2^{27}\)
\(\,\,\,\,\,\,2^{3^3}=134{,}217{,}728\)
The answer is \(134{,}217{,}728\)
\(\,\,\,\,\,\,2^{3^3}=2^{\left(3^3\right)}\)
\(\,\,\,\,\,\,2^{3^3}=2^{27}\)
\(\,\,\,\,\,\,2^{3^3}=134{,}217{,}728\)
The answer is \(134{,}217{,}728\)
\(\textbf{2)}\) \(3^{2^4}\)The answer is \(3^{16}=43{,}046{,}721\)
\(\,\,\,\,\,\,3^{2^4}=3^{\left(2^4\right)}\)
\(\,\,\,\,\,\,3^{2^4}=3^{16}\)
\(\,\,\,\,\,\,3^{2^4}=43{,}046{,}721\)
The answer is \(43{,}046{,}721\)
\(\,\,\,\,\,\,3^{2^4}=3^{\left(2^4\right)}\)
\(\,\,\,\,\,\,3^{2^4}=3^{16}\)
\(\,\,\,\,\,\,3^{2^4}=43{,}046{,}721\)
The answer is \(43{,}046{,}721\)
\(\textbf{3)}\) \(2^{1^{1^{3}}}\)The answer is \(2\)
\(\,\,\,\,\,\,2^{1^{1^{3}}}=2^{\left(1^{1^{3}}\right)}\)
\(\,\,\,\,\,\,2^{1^{1^{3}}}=2^{\left(1^{1}\right)}\)
\(\,\,\,\,\,\,2^{1^{1^{3}}}=2^1\)
\(\,\,\,\,\,\,2^{1^{1^{3}}}=2\)
The answer is \(2\)
\(\,\,\,\,\,\,2^{1^{1^{3}}}=2^{\left(1^{1^{3}}\right)}\)
\(\,\,\,\,\,\,2^{1^{1^{3}}}=2^{\left(1^{1}\right)}\)
\(\,\,\,\,\,\,2^{1^{1^{3}}}=2^1\)
\(\,\,\,\,\,\,2^{1^{1^{3}}}=2\)
The answer is \(2\)
\(\textbf{4)}\) \(5^{2^{2}}\)The answer is \(5^4=625\)
\(\,\,\,\,\,\,5^{2^2}=5^{\left(2^2\right)}\)
\(\,\,\,\,\,\,5^{2^2}=5^4\)
\(\,\,\,\,\,\,5^{2^2}=625\)
The answer is \(625\)
\(\,\,\,\,\,\,5^{2^2}=5^{\left(2^2\right)}\)
\(\,\,\,\,\,\,5^{2^2}=5^4\)
\(\,\,\,\,\,\,5^{2^2}=625\)
The answer is \(625\)
\(\textbf{5)}\) \(4^{2^{3}}\)The answer is \(4^8=65{,}536\)
\(\,\,\,\,\,\,4^{2^3}=4^{\left(2^3\right)}\)
\(\,\,\,\,\,\,4^{2^3}=4^8\)
\(\,\,\,\,\,\,4^{2^3}=65{,}536\)
The answer is \(65{,}536\)
\(\,\,\,\,\,\,4^{2^3}=4^{\left(2^3\right)}\)
\(\,\,\,\,\,\,4^{2^3}=4^8\)
\(\,\,\,\,\,\,4^{2^3}=65{,}536\)
The answer is \(65{,}536\)
\(\textbf{6)}\) \(3^{3^{2}}\)The answer is \(3^9=19{,}683\)
\(\,\,\,\,\,\,3^{3^2}=3^{\left(3^2\right)}\)
\(\,\,\,\,\,\,3^{3^2}=3^9\)
\(\,\,\,\,\,\,3^{3^2}=19{,}683\)
The answer is \(19{,}683\)
\(\,\,\,\,\,\,3^{3^2}=3^{\left(3^2\right)}\)
\(\,\,\,\,\,\,3^{3^2}=3^9\)
\(\,\,\,\,\,\,3^{3^2}=19{,}683\)
The answer is \(19{,}683\)
\(\textbf{7)}\) \(9=3^{4^{(\sqrt{2})^x}}\)The answer is \(x=-2\)
\(\,\,\,\,\,\,9=3^{4^{(\sqrt{2})^x}}\)
\(\,\,\,\,\,\,3^2=3^{4^{(\sqrt{2})^x}}\)
\(\,\,\,\,\,\,2=4^{(\sqrt{2})^x}\)
\(\,\,\,\,\,\,4^{1/2}=4^{(\sqrt{2})^x}\)
\(\,\,\,\,\,\,\frac12=(\sqrt{2})^x\)
\(\,\,\,\,\,\,2^{-1}=2^{x/2}\)
\(\,\,\,\,\,\,-1=\frac{x}{2}\)
\(\,\,\,\,\,\,x=-2\)
The answer is \(x=-2\)
\(\,\,\,\,\,\,9=3^{4^{(\sqrt{2})^x}}\)
\(\,\,\,\,\,\,3^2=3^{4^{(\sqrt{2})^x}}\)
\(\,\,\,\,\,\,2=4^{(\sqrt{2})^x}\)
\(\,\,\,\,\,\,4^{1/2}=4^{(\sqrt{2})^x}\)
\(\,\,\,\,\,\,\frac12=(\sqrt{2})^x\)
\(\,\,\,\,\,\,2^{-1}=2^{x/2}\)
\(\,\,\,\,\,\,-1=\frac{x}{2}\)
\(\,\,\,\,\,\,x=-2\)
The answer is \(x=-2\)
\(\textbf{8)}\) \(\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}\,\cdots}}}\)The answer is \(2\)
\(\,\,\,\,\,\,\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}\,\cdots}}}=x\)
\(\,\,\,\,\,\,x=(\sqrt{2})^x\)
\(\,\,\,\,\,\,x=(2^{1/2})^x\)
\(\,\,\,\,\,\,x=2^{x/2}\)
\(\,\,\,\,\,\,x=2\text{ and }x=4\text{ satisfy the fixed-point equation}\)
\(\,\,\,\,\,\,\text{The infinite tower generated by }\sqrt{2}\text{ converges to }2\)
The answer is \(2\)
\(\,\,\,\,\,\,\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}\,\cdots}}}=x\)
\(\,\,\,\,\,\,x=(\sqrt{2})^x\)
\(\,\,\,\,\,\,x=(2^{1/2})^x\)
\(\,\,\,\,\,\,x=2^{x/2}\)
\(\,\,\,\,\,\,x=2\text{ and }x=4\text{ satisfy the fixed-point equation}\)
\(\,\,\,\,\,\,\text{The infinite tower generated by }\sqrt{2}\text{ converges to }2\)
The answer is \(2\)
\(\textbf{9)}\) \(2^{2^4}\)The answer is \(65{,}536\)
\(\,\,\,\,\,\,2^{2^4}=2^{16}\)
\(\,\,\,\,\,\,2^{16}=65{,}536\)
The answer is \(65{,}536\)
\(\,\,\,\,\,\,2^{2^4}=2^{16}\)
\(\,\,\,\,\,\,2^{16}=65{,}536\)
The answer is \(65{,}536\)
\(\textbf{10)}\) \(10^{2^2}\)The answer is \(10{,}000\)
\(\,\,\,\,\,\,10^{2^2}=10^4\)
\(\,\,\,\,\,\,10^4=10{,}000\)
The answer is \(10{,}000\)
\(\,\,\,\,\,\,10^{2^2}=10^4\)
\(\,\,\,\,\,\,10^4=10{,}000\)
The answer is \(10{,}000\)
\(\textbf{11)}\) \(2^{3^2}\)The answer is \(512\)
\(\,\,\,\,\,\,2^{3^2}=2^9\)
\(\,\,\,\,\,\,2^9=512\)
The answer is \(512\)
\(\,\,\,\,\,\,2^{3^2}=2^9\)
\(\,\,\,\,\,\,2^9=512\)
The answer is \(512\)
\(\textbf{12)}\) \(4^{3^2}\)The answer is \(262{,}144\)
\(\,\,\,\,\,\,4^{3^2}=4^9\)
\(\,\,\,\,\,\,4^9=262{,}144\)
The answer is \(262{,}144\)
\(\,\,\,\,\,\,4^{3^2}=4^9\)
\(\,\,\,\,\,\,4^9=262{,}144\)
The answer is \(262{,}144\)
\(\textbf{13)}\) \(7^{2^2}\)The answer is \(2401\)
\(\,\,\,\,\,\,7^{2^2}=7^4\)
\(\,\,\,\,\,\,7^4=2401\)
The answer is \(2401\)
\(\,\,\,\,\,\,7^{2^2}=7^4\)
\(\,\,\,\,\,\,7^4=2401\)
The answer is \(2401\)
\(\textbf{14)}\) \(2^{2^{2^2}}\)The answer is \(65{,}536\)
\(\,\,\,\,\,\,2^{2^{2^2}}\)
\(\,\,\,\,\,\,2^{2^4}\)
\(\,\,\,\,\,\,2^{16}\)
\(\,\,\,\,\,\,65{,}536\)
The answer is \(65{,}536\)
\(\,\,\,\,\,\,2^{2^{2^2}}\)
\(\,\,\,\,\,\,2^{2^4}\)
\(\,\,\,\,\,\,2^{16}\)
\(\,\,\,\,\,\,65{,}536\)
The answer is \(65{,}536\)
\(\textbf{15)}\) Evaluate \(2^{3^2}\) and \((2^3)^2\).The answers are \(512\) and \(64\)
\(\,\,\,\,\,\,2^{3^2}=2^9\)
\(\,\,\,\,\,\,2^9=512\)
\(\,\,\,\,\,\,(2^3)^2=8^2\)
\(\,\,\,\,\,\,8^2=64\)
The answers are \(512\) and \(64\)
\(\,\,\,\,\,\,2^{3^2}=2^9\)
\(\,\,\,\,\,\,2^9=512\)
\(\,\,\,\,\,\,(2^3)^2=8^2\)
\(\,\,\,\,\,\,8^2=64\)
The answers are \(512\) and \(64\)
\(\textbf{16)}\) Solve: \(16=2^{2^x}\)The answer is \(x=2\)
\(\,\,\,\,\,\,16=2^{2^x}\)
\(\,\,\,\,\,\,2^4=2^{2^x}\)
\(\,\,\,\,\,\,4=2^x\)
\(\,\,\,\,\,\,2^2=2^x\)
\(\,\,\,\,\,\,x=2\)
The answer is \(x=2\)
\(\,\,\,\,\,\,16=2^{2^x}\)
\(\,\,\,\,\,\,2^4=2^{2^x}\)
\(\,\,\,\,\,\,4=2^x\)
\(\,\,\,\,\,\,2^2=2^x\)
\(\,\,\,\,\,\,x=2\)
The answer is \(x=2\)
\(\textbf{17)}\) Solve: \(81=3^{2^x}\)The answer is \(x=2\)
\(\,\,\,\,\,\,81=3^{2^x}\)
\(\,\,\,\,\,\,3^4=3^{2^x}\)
\(\,\,\,\,\,\,4=2^x\)
\(\,\,\,\,\,\,2^2=2^x\)
\(\,\,\,\,\,\,x=2\)
The answer is \(x=2\)
\(\,\,\,\,\,\,81=3^{2^x}\)
\(\,\,\,\,\,\,3^4=3^{2^x}\)
\(\,\,\,\,\,\,4=2^x\)
\(\,\,\,\,\,\,2^2=2^x\)
\(\,\,\,\,\,\,x=2\)
The answer is \(x=2\)
\(\textbf{18)}\) Solve: \(256=2^{2^{x+1}}\)The answer is \(x=2\)
\(\,\,\,\,\,\,256=2^{2^{x+1}}\)
\(\,\,\,\,\,\,2^8=2^{2^{x+1}}\)
\(\,\,\,\,\,\,8=2^{x+1}\)
\(\,\,\,\,\,\,2^3=2^{x+1}\)
\(\,\,\,\,\,\,3=x+1\)
\(\,\,\,\,\,\,x=2\)
The answer is \(x=2\)
\(\,\,\,\,\,\,256=2^{2^{x+1}}\)
\(\,\,\,\,\,\,2^8=2^{2^{x+1}}\)
\(\,\,\,\,\,\,8=2^{x+1}\)
\(\,\,\,\,\,\,2^3=2^{x+1}\)
\(\,\,\,\,\,\,3=x+1\)
\(\,\,\,\,\,\,x=2\)
The answer is \(x=2\)
\(\textbf{19)}\) Solve: \(27=3^{3^x}\)The answer is \(x=1\)
\(\,\,\,\,\,\,27=3^{3^x}\)
\(\,\,\,\,\,\,3^3=3^{3^x}\)
\(\,\,\,\,\,\,3=3^x\)
\(\,\,\,\,\,\,3^1=3^x\)
\(\,\,\,\,\,\,x=1\)
The answer is \(x=1\)
\(\,\,\,\,\,\,27=3^{3^x}\)
\(\,\,\,\,\,\,3^3=3^{3^x}\)
\(\,\,\,\,\,\,3=3^x\)
\(\,\,\,\,\,\,3^1=3^x\)
\(\,\,\,\,\,\,x=1\)
The answer is \(x=1\)
\(\textbf{20)}\) \(\sqrt[3]{3}^{\sqrt[3]{3}^{\sqrt[3]{3}^{\sqrt[3]{3}\,\cdots}}}\)The answer is \(3\)
\(\,\,\,\,\,\,\sqrt[3]{3}^{\sqrt[3]{3}^{\sqrt[3]{3}^{\sqrt[3]{3}\,\cdots}}}=x\)
\(\,\,\,\,\,\,x=(\sqrt[3]{3})^x\)
\(\,\,\,\,\,\,x=(3^{1/3})^x\)
\(\,\,\,\,\,\,x=3^{x/3}\)
\(\,\,\,\,\,\,3=3^{3/3}\)
\(\,\,\,\,\,\,3=3^1\)
\(\,\,\,\,\,\,\text{The convergent infinite tower approaches }3\)
The answer is \(3\)
\(\,\,\,\,\,\,\sqrt[3]{3}^{\sqrt[3]{3}^{\sqrt[3]{3}^{\sqrt[3]{3}\,\cdots}}}=x\)
\(\,\,\,\,\,\,x=(\sqrt[3]{3})^x\)
\(\,\,\,\,\,\,x=(3^{1/3})^x\)
\(\,\,\,\,\,\,x=3^{x/3}\)
\(\,\,\,\,\,\,3=3^{3/3}\)
\(\,\,\,\,\,\,3=3^1\)
\(\,\,\,\,\,\,\text{The convergent infinite tower approaches }3\)
The answer is \(3\)
See Related Pages\(\)
\(\bullet\text{ Intro to Exponents}\)
\(\,\,\,\,\,\,\,\,3^2=3 \times 3 =9\)
\(\bullet\text{ Negative Exponents}\)
\(\,\,\,\,\,\,\,\,3^{-2}=\frac{1}{9}\)
\(\bullet\text{ Rational Exponents}\)
\(\,\,\,\,\,\,\,\,\displaystyle x^{a/b}=\sqrt[b]{x^a}\)