Table of Contents
In arithmetic, powers and roots are ways of dealing with repeated multiplication and “undoing” that multiplication. They are closely related ideas, and both will appear again and again throughout later mathematics, so this chapter gives a unified introduction to them before the later subchapters go into details.
Powers: repeated multiplication
A power is a way to write repeated multiplication in a compact form.
Instead of writing
$$2 \times 2 \times 2,$$
we write
$$2^3.$$
Here:
- $2$ is called the base.
- $3$ is called the exponent (or power).
- $2^3$ is read as “two to the third power” or “two cubed”.
In general, for a positive integer $n$,
$$a^n = \underbrace{a \times a \times \cdots \times a}_{\text{$n$ factors of $a$}}.$$
Some common small powers are worth recognizing:
- $a^1 = a$ (multiplying $a$ just once gives $a$).
- $a^2$ is read as “$a$ squared”.
- $a^3$ is read as “$a$ cubed”.
Powers with larger exponents grow quickly. For example:
- $2^4 = 16$
- $2^5 = 32$
- $2^{10} = 1024$
Powers appear everywhere: in area and volume formulas, in scientific notation, in growth and decay problems, and later in algebra and beyond.
Square roots and cube roots
If a power answers “what do you get when you multiply something by itself several times?”, then a root answers “what number did you start with?”
Square roots
The square root of a number $x$ is a number $y$ such that
$$y^2 = x.$$
We write the (principal) square root of $x$ as $\sqrt{x}$.
So:
- $\sqrt{9} = 3$ because $3^2 = 9$.
- $\sqrt{16} = 4$ because $4^2 = 16$.
By convention, $\sqrt{x}$ means the non-negative square root. For instance, while $(-3)^2 = 9$ as well, we still write $\sqrt{9} = 3$, not $-3$.
For negative numbers, there is no real number whose square is negative, so expressions like $\sqrt{-4}$ do not represent real numbers in basic arithmetic (they will be revisited later using complex numbers).
Cube roots
The cube root of a number $x$ is a number $y$ such that
$$y^3 = x.$$
We write the cube root of $x$ as $\sqrt[3]{x}$.
Examples:
- $\sqrt[3]{8} = 2$ because $2^3 = 8$.
- $\sqrt[3]{27} = 3$ because $3^3 = 27$.
- $\sqrt[3]{-8} = -2$ because $(-2)^3 = -8$.
Unlike square roots, cube roots of negative numbers are defined within the real numbers, since a negative number cubed stays negative.
Higher roots
You can also talk about fourth roots, fifth roots, and so on. In general, an $n$th root of $x$ is a number $y$ such that
$$y^n = x.$$
We write:
- $n$th root of $x$ as $\sqrt[n]{x}$.
For example:
- $\sqrt[4]{16} = 2$ because $2^4 = 16$.
- $\sqrt[5]{32} = 2$ because $2^5 = 32$.
Roots with even indices (like square roots, fourth roots, etc.) and with odd indices (like cube roots, fifth roots, etc.) behave differently with negative numbers, as seen above.
Inverse relationship between powers and roots
Powers and roots are opposite operations in the same way that addition and subtraction are opposites, or multiplication and division are opposites.
For positive numbers and suitable exponents:
- If $y^2 = x$, then $\sqrt{x} = y$.
- If $y^3 = x$, then $\sqrt[3]{x} = y$.
- More generally, if $y^n = x$, then $\sqrt[n]{x} = y$ (when this makes sense as a real number).
In formulas, when everything is defined:
- $$\left(\sqrt[n]{x}\right)^n = x$$
- $$\sqrt[n]{x^n} = x \quad\text{(with attention to signs for even $n$).}$$
This inverse relationship is what allows you to use roots to “undo” powers and powers to “undo” roots.
Common uses of powers and roots in arithmetic
Even before formal algebra, powers and roots appear in many everyday contexts:
- Area and volume
- Area of a square of side length $s$: $s^2$.
- Volume of a cube of side length $s$: $s^3$.
- Finding a side from the area of a square uses a square root: if area is $A$, then side is $\sqrt{A}$.
- Scaling and growth
- Repeated percentage growth (like interest or population growth) is naturally expressed with powers.
- Halving or doubling repeatedly leads to powers of $\tfrac{1}{2}$ or $2$.
- Measurement conversions
- Converting between units sometimes involves squaring or cubing (for areas and volumes), and “undoing” that involves roots.
Later chapters on exponents, square roots, and laws of exponents will build on this overview and give detailed rules and methods for calculating and simplifying expressions with powers and roots.