- 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.
Radical (surd)
An irrational number written in radical form, such as $\sqrt{2}$ or $2\sqrt{3}$.
Radical sign
The symbol $\sqrt{\;}$ used to indicate a root.
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.
$25$ 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 $-\sqrt{x}$.
- 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.
- 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 $$a < \sqrt{x} < b.$$
- 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 $2.2<\sqrt{5}<2.3$
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.
- Only use "square both sides" when you are sure both sides are non-negative.
- Otherwise squaring can hide the sign and give misleading comparisons.
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 $x\ge 0$, $$\left(\sqrt{x}\right)^2=x.$$
$$(\sqrt{36})^2=36$$
$$(\sqrt{1001})^2=1001$$
The reverse expression is different: $\sqrt{x^2}=|x|$ because the principal square root is always non-negative.
Rule 2: Product Rule
For $a\ge 0$ and $b\ge 0$, $$\sqrt{ab}=\sqrt{a}\,\sqrt{b}.$$
$$\sqrt{4\cdot 9}=\sqrt{4}\sqrt{9}=2\cdot 3=6$$
- 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 $b>0$, $$\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}.$$
$$\sqrt{\tfrac12}=\frac{\sqrt{1}}{\sqrt{2}}=\frac{1}{\sqrt{2}}$$
Simplifying Radicals Is Like Simplifying Fractions
To simplify a radical, rewrite the radicand so it contains a perfect square factor, then use the product rule.
$$\sqrt{12}=\sqrt{4\cdot 3}=\sqrt{4}\sqrt{3}=2\sqrt{3}$$
The thinking is similar to simplifying fractions: factor first, then "pull out" what simplifies nicely.
Look for the largest perfect square that divides the radicand.
- For $12$, use $4$.
- For $72$, use $36$.
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 $5x\sqrt{2}$.
- 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.
- $2\sqrt{3}+5\sqrt{3}=7\sqrt{3}$
- $\sqrt{12}+\sqrt{3}=2\sqrt{3}+\sqrt{3}=3\sqrt{3}$
- $\sqrt{2}+\sqrt{3}$ cannot be simplified further (they are unlike surds)
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.
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.
- 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 $\sqrt{9}+\sqrt{16}$.
- 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.
