Powers and Roots
Integer & Fractional Exponents, and Simplification Rules
1. What Are Exponents?
Exponents provide a compact way of writing repeated multiplication of a number by itself.
In the expression $a^n$, the number $a$ is called the base and the superscript $n$ is called the exponent (also known as the power or index). The exponent tells us how many times the base appears as a factor in the product.
| Symbol | Name | Meaning |
|---|---|---|
| $a$ | Base | The number being multiplied |
| $n$ | Exponent (power / index) | How many times the base is used as a factor |
Reading aloud: $a^n$ is read as “$a$ to the power $n$” or “$a$ raised to the $n$th power.”
Special cases: $a^2$ is “$a$ squared”; $a^3$ is “$a$ cubed.”
Special cases: $a^2$ is “$a$ squared”; $a^3$ is “$a$ cubed.”
Example: $5^4 = 5 \times 5 \times 5 \times 5 = 625$
2. Positive Integer Exponents
When the exponent $n$ is a positive integer, the definition is simply:
$$a^n = \underbrace{a \times a \times a \times \cdots \times a}_{n \text{ factors}}$$
| Expression | Expanded Form | Value |
|---|---|---|
| $2^5$ | $2 \times 2 \times 2 \times 2 \times 2$ | $32$ |
| $(-3)^3$ | $(-3)\times(-3)\times(-3)$ | $-27$ |
| $(-2)^4$ | $(-2)\times(-2)\times(-2)\times(-2)$ | $16$ |
| $10^6$ | $10 \times 10 \times 10 \times 10 \times 10 \times 10$ | $1{}000{}000$ |
⚠️ Critical Distinction — Negative Signs:
$(-3)^2 = 9$ but $-3^2 = -(3^2) = -9$.
Without parentheses, the exponent applies only to the base $3$, and the negative sign is applied after the exponentiation.
$(-3)^2 = 9$ but $-3^2 = -(3^2) = -9$.
Without parentheses, the exponent applies only to the base $3$, and the negative sign is applied after the exponentiation.
Interactive: Visualising Exponential Growth
The graph below plots $y = 2^x$ for positive integer values of $x$. Notice how rapidly the values grow — this is the hallmark of exponential growth.
Interactive visualization — available in the full lesson