# Powers

*See also:*roots

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 \(f(x) = e^{-x}\).