Powers are mathematical operations that show how many times a number is to be multiplied by itself. The standard way of writing a power is \(a^{\large{n}}\) where \(a\) is the base and \(n\) is the exponent.

\[a^n = a \cdot a \cdot a \cdot a… \cdot a \quad \text{(for n times)}\]

where \[ a \in \mathbb{R}, \quad n \in \mathbb{Z}^+ \]

From a geometric perspective, when considering a positive number \( a \), it can be observed that the expressions \( a^2 \) and \( a^3 \) respectively denote the area of a square with its side length being \( a \) and the volume of a cube with its edge length being \( a \). Here is a simple matrix that illustrates the process of expanding a base \(a\) to an exponent \(n\). The next paragraph explains the properties that underlie the calculations.

\begin{array}{c|ccccc} & a^{-2} & a^{-1} & a^0 & a^1 & a^2 \\[0.6em] \hline -2 & \frac{1}{4} & -\frac{1}{2} & 1 & -2 & 4 \\[0.6em] -1 & 1 & -1 & 1 & -1 & 1 \\[0.6em] 0 & 0 & 0 & — & 0 & 0 \\[0.6em] 1 & 1 & 1 & 1 & 1 & 1\\[0.6em] 2 & \frac{1}{4} & \frac{1}{2} & 1 & 2 & 4 \ \end{array}


Properties

Without dwelling on the definitions and the most elementary concepts, let’s list the fundamental rules applicable for manipulating expressions containing powers within them.

  • Raising any base not equal to zero to an exponent of zero always results in \(1\). The value of \(a\) is restricted because the operation \(0^0\) is meaningless and considered an indeterminate form: \[a^0 = 1 \ \text{if} \ a\neq 0 \]

  • Any power of zero always results in zero because it corresponds to the product of \( n \) zeros, where \( n \) is the exponent: \[0^n = 0\]

  • The product of two or more powers with base \(a\) is a power with the same base \(a\), and the exponent equals the sum of the exponents: \[a^n \cdot a^m = a^{n+m}\]

  • The quotient of two or more powers with the same base \(a\) is a power with the same base \(a\) and exponent equal to the difference between the exponents:\[ \frac{a^n}{a^m} = a^{n-m}\]

  • The product of powers with different bases \(a\) and \(b\), and the same exponent \(n\), is a new power with a base equal to the product of the original bases and exponent equal to the exponent: \[a^{n} \cdot b^n = (ab)^n\]

  • The power of a power is a power that has the same base and exponent equal to the product of the exponents: \[(a^m)^n = a^{m \cdot n} \]

  • When a base \(a\) is raised to a negative power \(n\), we obtain the reciprocal of \(a\) raised to the positive power \(n\): \[a^{-n} = \frac{1}{a^n} \ \text{with} \ a \neq 0 \ \text{and} \ n > 0 \]

  • When the exponent is a rational number of the form \( \large{\frac{m}{n}} \), we have: \[a^{\frac{m}{n}} = \sqrt[n]{a^m} \quad \text{where} \ m, n \in \mathbb{N} \ \text{and} \ n \neq 0 \]


Powers and exponentials

Sometimes, people mistakenly mix up the concepts of power and exponential. Power is the arithmetic operation of multiplying a base \(a\) by itself for \(n\) times, where \(n\) is the exponent. Exponential, on the other hand, refers to the exponential function, which is represented by \(f(x) = e^x\).

Graphical representation
of the function \(f(x) = e^x\).

Graphical representation
of the function \(f(x) = e^{-x}\).