An exponential function is a function of the form $$y = \Tblue{k} \Torange{a}^{x}.$$ The number $\Torange{a}$ is positive.
The computation without a calculator is difficult for a general $x$. However, it can be done when $x$ is rational.
• When $x$ is an integer: $y$ is the usual $\Torange{a}$ to the power of $x$. $$2^{\Tred{1}} = 2,\; 2^{\Tred{2}} = 2\times2 = 4,\; 2^{\Tred{3}} = 2^2\times 2 = 8,\; 2^{\Tred{4}} = 16.$$
• When $x$ is the reciprocal of an integer: $y$ is the $x$-th root of $\Torange{a}$, i.e. the number that is $\Torange{a}$ when raised to the power $x$ $$2^{1/\Tred{2}} = \sqrt{2} \simeq 1.41,\; 2^{1/\Tred{3}} = \sqrt[\Tred{3}]{2}\simeq 1.26,\; 2^{1/\Tred{4}} = \sqrt[\Tred{4}]{2}\simeq 1.19.$$
• When $x$ is rational ($p/q$): $y$ is the $q$-th root of $\Torange{a}$ raised to the power $p$ $$2^{\Tblue{3}/\Tred{2}} = (\sqrt{2})^\Tblue{3} \simeq (1.41)^\Tblue{3}\simeq 1.58,\; 2^{\Tblue{2}/\Tred{3}} = (\sqrt[\Tred{3}]{2})^\Tblue{2}\simeq 2.83$$
Graph of the exponential function $y = 2^x$