Radicals: definition, existence conditions, fundamental properties, simplification, operations and rationalisation. Examples and step-by-step solved exercises.
Table of Contents
- Definition of a radical
- Existence conditions
- Fundamental properties
- Simplifying radicals
- Multiplication and division
- Addition and subtraction
- Powers of radicals
- Rationalising the denominator
- Radicals with variables
- Radical equations
Definition of a radical
The nth root of a real number \(a\) is the number \(b\) such that raising it to the nth power gives back \(a\).
Definition
Given \( n \in \mathbb{N} \), \( n \geq 2 \) and \( a \in \mathbb{R} \), the nth root of \( a \) is defined as the real number \( b \) satisfying: \[ b = \sqrt[n]{a} \quad \Longleftrightarrow \quad b^n = a \]
The number \( n \) is the index of the radical, and \( a \) is the radicand.
Square Root
By convention, when \( n = 2 \) the index is omitted:
\[ \sqrt{a} = \sqrt[2]{a} \]
The square root always returns the principal non-negative value and is defined only for \( a \geq 0 \). The following identity is important:
\[ \sqrt{a^2} = |a| \] Warning. In general \( \sqrt{a^2} \neq a \). For instance, \( \sqrt{(-3)^2} = \sqrt{9} = 3 \neq -3 \).
nth Root: parity of the index
| Index \( n \) | Radicand \( a \) | Result |
|---|---|---|
| Even | \( a > 0 \) | a unique positive real value exists (principal root) |
| Even | \( a = 0 \) | \( \sqrt[n]{0} = 0 \) |
| Even | \( a < 0 \) | does not exist in \( \mathbb{R} \) |
| Odd | any \( a \in \mathbb{R} \) | a unique real value exists, with the same sign as \( a \) |
Examples
\( \sqrt[3]{-8} = -2 \) since \( (-2)^3 = -8 \)
\( \sqrt[4]{16} = 2 \) (principal root)
\( \sqrt[5]{-32} = -2 \) since \( (-2)^5 = -32 \)
Existence conditions
A radical is a real number only when the radicand satisfies the following conditions, which depend on the parity of the index.
Existence condition
\[ \sqrt[n]{a} \in \mathbb{R} \quad \Longleftrightarrow \quad \begin{cases} a \geq 0 & \text{if } n \text{ is even} \\ a \in \mathbb{R} & \text{if } n \text{ is odd} \end{cases} \] Examples
\( \sqrt{x-3} \) exists \(\iff\) \( x-3 \geq 0 \) \(\iff\) \( x \geq 3 \)
\( \sqrt[3]{x-3} \) exists for every \( x \in \mathbb{R} \)
\( \sqrt{x^2-4} \) exists \(\iff\) \( x \leq -2 \) or \( x \geq 2 \)
Fundamental properties
The following properties hold whenever all expressions are defined in the real numbers (in particular, for even indices, all radicands must be non-negative).
| Property | Formula |
|---|---|
| Radical of a power | \( \sqrt[n]{a^m} = a^{m/n} \) (with \( a \geq 0 \) if \( n \) is even) |
| Power of a radical | \( (\sqrt[n]{a})^m = \sqrt[n]{a^m} \) |
| Nested radical | \( \sqrt[m]{\sqrt[n]{a}} = \sqrt[mn]{a} \) |
| Reduction to a common index | \( \sqrt[n]{a} = \sqrt[kn]{a^k} \) for \( k \in \mathbb{N}, k \geq 1 \) |
| Index simplification | \( \sqrt[kn]{a^k} = \sqrt[n]{a} \) |
Connection with fractional exponents
\[ \sqrt[n]{a^m} = a^{m/n} \]
Simplifying radicals
A radical is in simplified form when the radicand contains no factors that are perfect powers of the index — that is, no factors that can be extracted as whole numbers.
Method for simplification
- Factor the radicand into primes (or into factors with explicit exponents).
- Write each exponent as a multiple of \( n \) plus a remainder \( r \) with \( 0 \leq r < n \).
- Extract from the radical any part whose exponent is a multiple of the index.
\[ \sqrt[n]{a^{qn+r}} = a^q \sqrt[n]{a^r}, \quad 0 \leq r < n \quad (a \geq 0 \text{ if } n \text{ even}) \] Examples
\( \sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2} \)
\( \sqrt[3]{54} = \sqrt[3]{27 \cdot 2} = 3\sqrt[3]{2} \)
\( \sqrt{x^5} = x^2 \sqrt{x} \) for \( x \geq 0 \)
\( \sqrt[3]{a^8} = a^2 \sqrt[3]{a^2} \)
Reduction to a common index
To combine radicals with different indices, use the least common multiple (LCM) of the indices.
Example
\( \sqrt{2} = \sqrt[6]{2^3} = \sqrt[6]{8} \)
\( \sqrt[3]{3} = \sqrt[6]{3^2} = \sqrt[6]{9} \)
Multiplication and division
Properties (for defined expressions)
\[ \sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab}, \qquad \frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}} \quad (b > 0) \] Warning. These properties hold only when all radicands satisfy the existence conditions. Examples
\( \sqrt{3} \cdot \sqrt{12} = \sqrt{36} = 6 \)
\( \sqrt[3]{4} \cdot \sqrt[3]{2} = \sqrt[3]{8} = 2 \)
\( \frac{\sqrt{50}}{\sqrt{2}} = \sqrt{25} = 5 \)
\( \sqrt{2} \cdot \sqrt[3]{2} = \sqrt[6]{2^5} = \sqrt[6]{32} \)
Addition and subtraction
Only like radicals — those sharing the same index and the same radicand — can be added or subtracted.
Like radicals
\( p\sqrt[n]{a} \pm q\sqrt[n]{a} = (p \pm q)\sqrt[n]{a} \) Examples
\( 3\sqrt{2} + 5\sqrt{2} = 8\sqrt{2} \)
\( \sqrt{12} + \sqrt{27} = 2\sqrt{3} + 3\sqrt{3} = 5\sqrt{3} \)
\( \sqrt{8} - \sqrt{2} + \sqrt{18} = 2\sqrt{2} - \sqrt{2} + 3\sqrt{2} = 4\sqrt{2} \)
Powers of radicals
\[ (\sqrt[n]{a})^m = \sqrt[n]{a^m} = a^{m/n} \quad (a \geq 0 \text{ if } n \text{ even}) \]
Square of a binomial involving radicals
\[ (\sqrt{a} + \sqrt{b})^2 = a + 2\sqrt{ab} + b \]
\[ (\sqrt{a} - \sqrt{b})^2 = a - 2\sqrt{ab} + b \]
Product of conjugate expressions
\[ (\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b}) = a - b \]
Rationalising the denominator
Rationalising the denominator means rewriting a fraction so that no radicals appear in the denominator, by multiplying both numerator and denominator by a suitable factor.
Case 1 — Denominator with a single radical
\[ \frac{b}{\sqrt[n]{a}} = \frac{b \cdot \sqrt[n]{a^{n-1}}}{a} \quad (a \geq 0 \text{ if } n \text{ even}) \] Examples
\[ \frac{3}{\sqrt{5}} = \frac{3\sqrt{5}}{5} \]
\[ \frac{1}{\sqrt[3]{2}} = \frac{\sqrt[3]{4}}{2} \]
Case 2 — Binomial denominator with square roots
\[ \frac{c}{\sqrt{a} \pm \sqrt{b}} = \frac{c(\sqrt{a} \mp \sqrt{b})}{a - b} \] Examples
\[ \frac{4}{\sqrt{3} + \sqrt{2}} = \frac{4(\sqrt{3} - \sqrt{2})}{3-2} = 4\sqrt{3} - 4\sqrt{2} \]
\[ \frac{1}{1 + \sqrt{5}} = \frac{\sqrt{5} - 1}{5-1} = \frac{\sqrt{5}-1}{4} \]
Case 3 — Denominator with cube roots (sum or difference)
The sum and difference of cubes identities are used:
\[ x^3 + y^3 = (x+y)(x^2 - xy + y^2) \qquad x^3 - y^3 = (x-y)(x^2 + xy + y^2) \]
For \( \frac{1}{\sqrt[3]{a} - \sqrt[3]{b}} \) (setting \( x = \sqrt[3]{a} \), \( y = \sqrt[3]{b} \)):
\[ \frac{1}{\sqrt[3]{a} - \sqrt[3]{b}} = \frac{\sqrt[3]{a^2} + \sqrt[3]{ab} + \sqrt[3]{b^2}}{a - b} \] Example
\[ \frac{1}{\sqrt[3]{2} - 1} = \frac{\sqrt[3]{4} + \sqrt[3]{2} + 1}{2 - 1} = \sqrt[3]{4} + \sqrt[3]{2} + 1 \]
Radicals with variables
Absolute value in simplification
For an even index (\( n = 2k \)): \( \sqrt[2k]{x^{2k}} = |x| \)
For an odd index: \( \sqrt[2k+1]{x^{2k+1}} = x \) Examples
\( \sqrt{x^2} = |x| \)
\( \sqrt[4]{x^4} = |x| \)
\( \sqrt{x^6} = |x^3| \)
\( \sqrt[3]{x^3} = x \)
Domain of expressions involving several radicals
The domain is the intersection of the existence conditions of all radicals present.
Example
\( f(x) = \sqrt{x+2} + \sqrt{4-x} \)
Domain: \( x \geq -2 \) and \( x \leq 4 \) \(\Rightarrow\) \( [-2, 4] \)
Radical equations
To solve a radical equation, follow these steps:
- Determine the domain (existence conditions of all radicals).
- Isolate one radical (if possible).
- Raise both sides to the appropriate power.
- Solve the resulting algebraic equation.
- Check each candidate solution in the original equation and verify it belongs to the domain (to discard any extraneous solutions).
Warning. Raising both sides to a power may introduce extraneous solutions. Verification is mandatory.
Example — even index
\( \sqrt{2x-1} = x-2 \)
Domain: \( x \geq \frac{1}{2} \) and \( x-2 \geq 0 \) \(\Rightarrow\) \( x \geq 2 \).
Squaring both sides: \( 2x-1 = (x-2)^2 \Rightarrow x^2 - 6x + 5 = 0 \Rightarrow x=1 \) or \( x=5 \).
Check: \( x=1 \) does not belong to the domain → extraneous solution.
\( x=5 \): \( \sqrt{10-1} = 3 \) and \( 5-2=3 \) → verified.
Solution: \( x=5 \)
Example — two radicals
\( \sqrt{x+5} - \sqrt{x} = 1 \)
Domain: \( x \geq 0 \).
Isolate: \( \sqrt{x+5} = \sqrt{x} + 1 \).
Square both sides: \( x+5 = x + 2\sqrt{x} + 1 \Rightarrow 4 = 2\sqrt{x} \Rightarrow x=4 \).
Check: \( \sqrt{9} - \sqrt{4} = 3-2=1 \) → correct.
Solution: \( x=4 \)