Radicals (also called surds) are a way of writing many irrational numbers exactly.
In this topic you will learn to recognize rational and irrational numbers, use key rules of radicals, simplify surd expressions, and estimate surds without a calculator.
What Radicals Represent
Many measurements and calculations produce numbers like $\sqrt{2}$ or $\sqrt{5}$.
These cannot be written exactly as terminating or repeating decimals, but they can be written exactly using surd notation.
Definition
Radical (surd)
An irrational number written in radical form, such as $\sqrt{2}$ or $2\sqrt{3}$.
Definition
Radical sign
The symbol $\sqrt{\;}$ used to indicate a root.
Definition
Radicand
The number or expression inside the radical sign, for example the $5$ in $\sqrt{5}$.
A square root of a positive number $x$ is a number which, when multiplied by itself, gives $x$.
Any positive number has two square roots, one positive and one negative.
Example
$25$ has square roots $5$ and $-5$.
In most school mathematics, "the square root of $x$" means the principal square root, the positive one, written $\sqrt{x}$.
The negative square root is written $-\sqrt{x}$.
Note
Using the radical symbol is the only way to write many square roots exactly.
For example, $\sqrt{5}$ cannot be written as a terminating or repeating decimal.
Rational And Irrational Numbers Describe Different Decimal Behaviors
A rational number is a number that can be written as a fraction $\frac{p}{q}$ where $p,q\in\mathbb{Z}$ and $q\neq 0$.
Rational numbers have decimal expansions that terminate (end) or repeat.
An irrational number cannot be written as $\frac{p}{q}$.
Irrational numbers have decimal expansions that do not terminate and do not repeat (like $\pi$).
Many square roots are irrational.
For example, $\sqrt{2}$ is irrational.
Common Mistake
A number can be written with a radical sign and still be rational.
For example, $\sqrt{9}=3$, so it is rational.
Approximating Radicals Lets You Compare And Estimate
You often need to compare a radical with a fraction (or another radical) without a calculator, for example deciding whether $\sqrt{5}$ is larger or smaller than $\frac{5}{2}$.
Two powerful strategies are (1) bounding using perfect squares, and (2) squaring to avoid decimals.
Bounding By Perfect Squares
If $a^2 < x < b^2$ with $a,b>0$, then $$a < \sqrt{x} < b.$$
Example
Since $2^2=4<5<9=3^2$, we know $$2<\sqrt{5}<3$$
To refine the estimate, test nearby decimals:
$2.2^2=4.84$ (a bit small)
$2.3^2=5.29$ (a bit large)
So $2.2<\sqrt{5}<2.3$
Example question
Compare $\sqrt{5}$ and $\frac{5}{2}$ without a calculator.
Solution
We know $\frac{5}{2}=2.5$, and we bounded $\sqrt{5}$ between $2.2$ and $2.3$. Therefore,
$$\sqrt{5}<\frac{5}{2}$$
Comparing By Squaring (Exact Method)
If both quantities are non-negative, you can square both sides without changing the inequality direction: $$\sqrt{x} \;?\; \frac{p}{q}\quad \Longleftrightarrow\quad x \;?\; \left(\frac{p}{q}\right)^2$$
This keeps the comparison exact, with no rounding.
Common Mistake
Only use "square both sides" when you are sure both sides are non-negative.
Otherwise squaring can hide the sign and give misleading comparisons.
Example question
Compare $\sqrt{2}$ and $\frac{7}{5}$.
Solution
Square the fraction: $$\left(\frac{7}{5}\right)^2=\frac{49}{25}=1.96$$
Because $2>1.96$, it follows that $$\sqrt{2}>\frac{7}{5}$$
Rules Of Radicals Follow Consistent Patterns
Radicals "behave" according to rules that connect closely to algebra and exponent laws.
These rules are the foundation for simplifying and operating with surds.
Rule 1: Squaring A Square Root Returns The Radicand
For $x\ge 0$, $$\left(\sqrt{x}\right)^2=x.$$
Example
$$(\sqrt{36})^2=36$$
$$(\sqrt{1001})^2=1001$$
Note
The reverse expression is different: $\sqrt{x^2}=|x|$ because the principal square root is alwaysnon-negative.
Rule 2: Product Rule
For $a\ge 0$ and $b\ge 0$, $$\sqrt{ab}=\sqrt{a}\,\sqrt{b}.$$
Example
$$\sqrt{4\cdot 9}=\sqrt{4}\sqrt{9}=2\cdot 3=6$$
Common Mistake
There is no similar rule for addition and subtraction.
In general, $$\sqrt{a+b}\ne \sqrt{a}+\sqrt{b}$$
For instance, $\sqrt{9+16}=5$ but $\sqrt{9}+\sqrt{16}=7$
Rule 3: Quotient Rule
For $a\ge 0$ and $b>0$, $$\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}.$$
The thinking is similar to simplifying fractions: factor first, then "pull out" what simplifies nicely.
Tip
Look for the largest perfect square that divides the radicand.
For $12$, use $4$.
For $72$, use $36$.
Hint
Simplifying With Variables
The same method works with algebraic expressions, but you must handle squares carefully: $$\sqrt{50x^2}=\sqrt{25\cdot 2\cdot x^2}=5\sqrt{2}\,|x|$$
If you are told $x\ge 0$, then $|x|=x$ and the expression becomes $5x\sqrt{2}$.
Common Mistake
A very common mistake is writing $\sqrt{x^2}=x$ without conditions.
The correct identity is: $$\sqrt{x^2}=|x|$$
Adding, Subtracting, And Multiplying Surds Uses Algebra Ideas
Surds combine like algebraic terms: you can only add or subtract like surds, meaning they have the same simplified radical part.
When multiplying, you distribute as usual and then simplify radicals.
Example
$2\sqrt{3}+5\sqrt{3}=7\sqrt{3}$
$\sqrt{12}+\sqrt{3}=2\sqrt{3}+\sqrt{3}=3\sqrt{3}$
$\sqrt{2}+\sqrt{3}$ cannot be simplified further (they are unlike surds)
Example question
Simplify $(3+\sqrt{2})(3-\sqrt{2})$.
Solution
Use difference of squares: $$(3+\sqrt{2})(3-\sqrt{2})=3^2-(\sqrt{2})^2=9-2=7$$
This is an important idea: irrational numbers can combine to make a rational number.
Case study
Radicals In Real Contexts: A Tsunami Speed Model
Square roots appear in models of real systems.
One simplified relationship for tsunami speed is $$s=\sqrt{gd}$$ where:
$s$ is speed (m/s)
$g\approx 9.8\,\text{m/s}^2$ is gravitational acceleration
$d$ is ocean depth (m)
Because $s$ depends on $\sqrt{d}$, speed increases more slowly than depth. In particular, multiplying depth by $4$ multiplies speed by $2$.
If depth increases from $2500\,\text{m}$ to $10000\,\text{m}$ (a factor of $4$), then $$\frac{s_2}{s_1}=\frac{\sqrt{g\cdot 10000}}{\sqrt{g\cdot 2500}}=\sqrt{\frac{10000}{2500}}=\sqrt{4}=2$$
So the tsunami speed approximately doubles.
Exact Form Versus Decimal Approximation Changes How You Operate
Writing $\sqrt{5}$ keeps the value exact.
Writing $\sqrt{5}\approx 2.236$ is useful for measurement, but it introduces rounding error.
Exact form is best for algebraic manipulation (simplifying, comparing exactly, proving).
Decimal form is best for estimates and practical calculations.
Active recall
Simplify $\sqrt{18}$.
Decide whether $\sqrt{3}$ is greater or less than $\frac{433}{250}$ without using a calculator.
Explain why $\sqrt{9+16}$ is not equal to $\sqrt{9}+\sqrt{16}$.
Exam technique
Check whether the radicand is a perfect square (then the result is rational).
Simplify radicals before adding or subtracting, to identify like surds.
Use $\sqrt{ab}=\sqrt{a}\sqrt{b}$ and $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$ only with non-negative radicands.
Remember $\sqrt{x^2}=|x|$.
When comparing, prefer exact methods (squaring, bounding by perfect squares) before rounding.