Radicals: Definition, Properties and Examples

Radicals arise as the inverse operations of powers and allow us to represent square roots, cube roots and, more generally, \(n\)th roots. Their study requires particular care, however, since the meaning of a root depends essentially on the index of the radical and on the sign of the radicand.

On this page we introduce a rigorous definition of radicals over the real numbers, drawing a precise distinction between the case of even index and the case of odd index. We shall see, in particular, that when the index is even, the symbol \(\sqrt[n]{a}\) does not denote all the solutions of the equation \(x^n=a\), but only the non-negative principal root, when it exists.

We shall then study the conditions of existence, the fundamental properties of radicals, simplification, operations on radicals, and rationalisation of the denominator. The final part will be devoted to radicals involving variables and to a first treatment of irrational equations, in which it is essential to determine the domain and to check the solutions obtained.

Contents

• Definition of a radical
• Conditions of existence
• Fundamental properties
• Simplification of radicals
• Multiplication and division
• Addition and subtraction
• Powers of radicals
• Rationalisation of the denominator
• Radicals involving variables
• Irrational equations

Definition of a radical

The \(n\)th radical of a real number \(a\) is the operation that recovers, when possible, a number whose \(n\)th power equals \(a\). Over the real numbers, however, the definition depends on the parity of the index \(n\).

Let \(n\in\mathbb{N}\), with \(n\geq 2\).

If \(n\) is even and \(a\geq0\), we define \(\sqrt[n]{a}\) to be the unique real number \(b\geq0\) such that

\[ b^n=a. \]

In symbols:

\[ b=\sqrt[n]{a} \quad \Longleftrightarrow \quad b^n=a,\quad b\geq0. \]

If instead \(n\) is odd and \(a\in\mathbb{R}\), we define \(\sqrt[n]{a}\) to be the unique real number \(b\) such that

\[ b^n=a. \]

In symbols:

\[ b=\sqrt[n]{a} \quad \Longleftrightarrow \quad b^n=a. \]

The number \(n\) is called the index of the radical, while the number \(a\) is called the radicand.

The distinction between an even index and an odd index is fundamental. If the index is even, the radicand must be non-negative, and the radical always denotes the non-negative principal root. If the index is odd, on the other hand, the radical is defined for every real radicand and preserves the sign of the radicand.

Square roots

The most important case is that of the square root. When \(n=2\), the index is omitted:

\[ \sqrt{a}=\sqrt[2]{a}. \]

The square root is defined over the real numbers only for \(a\geq0\), and it always returns the non-negative principal value.

In particular, for every \(a\in\mathbb{R}\), the identity

\[ \sqrt{a^2}=|a|. \]

holds. One must therefore not confuse \(\sqrt{a^2}\) with \(a\). In general, indeed,

\[ \sqrt{a^2}\neq a. \]

For instance:

\[ \sqrt{(-3)^2}=\sqrt{9}=3\neq -3. \]

\(n\)th roots and the parity of the index

Let us consider a few examples.

\[ \sqrt[3]{-8}=-2, \]

since

\[ (-2)^3=-8. \]

Furthermore

\[ \sqrt[4]{16}=2. \]

Indeed \(2^4=16\), but the radical \(\sqrt[4]{16}\) denotes the non-negative principal root, not the value \(-2\), even though \((-2)^4=16\).

Finally:

\[ \sqrt[5]{-32}=-2, \]

since

\[ (-2)^5=-32. \]

Conditions of existence

The conditions of existence of a radical establish the values of the radicand for which the radical is defined over the real numbers. Here too one must distinguish between an even index and an odd index.

If the index \(n\) is even, the radical

\[ \sqrt[n]{a} \]

exists over the real numbers if and only if

\[ a\geq0. \]

If instead the index \(n\) is odd, the radical

\[ \sqrt[n]{a} \]

exists for every real value of \(a\).

We may summarise the conditions of existence as follows:

\[ \sqrt[n]{a}\in\mathbb{R} \quad\Longleftrightarrow\quad \begin{cases} a\geq0 & \text{if } n \text{ is even},\\ a\in\mathbb{R} & \text{if } n \text{ is odd}. \end{cases} \]

Let us consider a few examples.

The radical

\[ \sqrt{x-3} \]

has even index, so it exists if and only if

\[ x-3\geq0. \]

Hence the condition of existence is

\[ x\geq3. \]

The radical

\[ \sqrt[3]{x-3} \]

has odd index instead, so it exists for every \(x\in\mathbb{R}\).

Finally, the radical

\[ \sqrt{x^2-4} \]

has even index, so we must impose

\[ x^2-4\geq0. \]

Solving the inequality, we obtain

\[ x\leq -2 \quad \text{or} \quad x\geq2. \]

Thus the radical \(\sqrt{x^2-4}\) exists for

\[ x\in(-\infty,-2]\cup[2,+\infty). \]

Fundamental properties

The properties of radicals allow us to transform and simplify expressions containing roots. Over the real numbers, however, these properties must be applied while respecting the conditions of existence and the convention of the principal root.

In particular, when the index is even, every radical involved must be defined over the real numbers, and the value of the radical is always non-negative. For this reason, certain formulae that appear straightforward call for care regarding the signs of the quantities involved.

Product and quotient

If two radicals share the same index, they may be multiplied or divided by combining the product or quotient under a single radical, provided all the expressions involved are defined.

For instance:

\[ \sqrt{3}\cdot\sqrt{12}=\sqrt{36}=6. \]

Likewise:

\[ \sqrt[3]{4}\cdot\sqrt[3]{2}=\sqrt[3]{8}=2. \]

For the quotient:

\[ \frac{\sqrt{50}}{\sqrt{2}}=\sqrt{\frac{50}{2}}=\sqrt{25}=5. \]

In the case of an even index, however, one must bear in mind that the radicands must be non-negative and that, in a quotient, the denominator must be strictly positive.

Powers of radicals

If the radical \(\sqrt[n]{a}\) is defined, then it may be raised to a natural-number power:

\[ \left(\sqrt[n]{a}\right)^m=\sqrt[n]{a^m}. \]

When \(a\geq0\), this property is linked to the notation with a rational exponent:

\[ \sqrt[n]{a^m}=a^{\frac{m}{n}}. \]

This notation is particularly useful in algebraic calculation, but it must be used with care when working over the real numbers with radicands that may take negative values.

Simplifying perfect powers

One of the most important identities is the following:

\[ \sqrt[2k]{a^{2k}}=|a|. \]

The absolute value is necessary because a root with even index always returns the non-negative principal value.

For instance:

\[ \sqrt{x^2}=|x|. \]

Likewise:

\[ \sqrt[4]{x^4}=|x|. \]

If instead the index is odd, no absolute value appears:

\[ \sqrt[2k+1]{a^{2k+1}}=a. \]

For instance:

\[ \sqrt[3]{x^3}=x. \]

Reduction to a common index

Reduction to a common index allows radicals with different indices to be rewritten as radicals sharing a common index. This transformation is particularly useful when radicals must be multiplied, divided, or compared.

If \(a\geq0\), then:

\[ \sqrt[n]{a}=\sqrt[kn]{a^k}, \qquad k\in\mathbb{N}^*. \]

For instance:

\[ \sqrt{2}=\sqrt[6]{2^3}=\sqrt[6]{8}. \]

Furthermore:

\[ \sqrt[3]{3}=\sqrt[6]{3^2}=\sqrt[6]{9}. \]

This property must be applied with caution when the radicand is negative. Indeed, when one passes to an even index, the radical always represents a non-negative principal root, and the sign may change.

For instance:

\[ \sqrt[3]{-8}=-2, \]

whereas

\[ \sqrt[6]{(-8)^2}=\sqrt[6]{64}=2. \]

Hence, in general, one cannot reduce the index without checking the conditions and the sign of the radicand.

Simplification of radicals

To simplify a radical means to rewrite it in an equivalent form in which the radicand no longer contains factors that are perfect powers of the index.

The idea is to separate, whenever possible, the powers that can be extracted from the radical from those that must remain under the root.

If \(a\geq0\), \(n\geq2\), \(q\in\mathbb{N}\), \(0\leq r<n\) and \(qn+r\geq1\), then

\[ \sqrt[n]{a^{qn+r}}=a^q\sqrt[n]{a^r}. \]

This formula expresses the general principle of simplification: exponents that are multiples of the index can be brought outside the radical, while the remaining powers stay in the radicand.

Method of simplification

A radical may be simplified as follows.

1. Factorise the radicand into prime factors, or into factors with exponents.
2. Write each exponent as a multiple of the index plus a remainder.
3. Bring the factors corresponding to perfect powers of the index outside the radical.

Let us consider a few examples.

\[ \sqrt{72}=\sqrt{36\cdot2}=6\sqrt{2}. \]

Indeed \(36\) is a perfect square and can be extracted from the square root.

Furthermore:

\[ \sqrt[3]{54}=\sqrt[3]{27\cdot2}=3\sqrt[3]{2}. \]

In this case \(27=3^3\) is a perfect power of index \(3\).

Simplification with variables

When the radicand contains real variables, particular attention must be paid to the sign. In particular, with even indices it may be necessary to introduce the absolute value.

For instance, for every \(x\in\mathbb{R}\) we have

\[ \sqrt{x^2}=|x|. \]

If instead we know that \(x\geq0\), then we may write

\[ \sqrt{x^5}=\sqrt{x^4\cdot x}=x^2\sqrt{x}. \]

The condition \(x\geq0\) is necessary for the radical \(\sqrt{x^5}\) to be defined over the real numbers.

For an odd index, on the other hand, no absolute value need be introduced. For instance:

\[ \sqrt[3]{a^8}=\sqrt[3]{a^6\cdot a^2}=a^2\sqrt[3]{a^2}. \]

Multiplication and division

To multiply or divide radicals with the same index, one uses the property of the product or of the quotient, always respecting the conditions of existence.

If the radicals share the same index, then:

\[ \sqrt[n]{a}\cdot\sqrt[n]{b}=\sqrt[n]{ab}. \]

For instance:

\[ \sqrt{3}\cdot\sqrt{12}=\sqrt{36}=6. \]

Likewise:

\[ \sqrt[3]{4}\cdot\sqrt[3]{2}=\sqrt[3]{8}=2. \]

For the quotient we have:

\[ \frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}, \]

provided the radical in the denominator is defined and non-zero. In particular, if the index is even, one must require \(b>0\); if the index is odd, it suffices to require \(b\neq0\).

For instance:

\[ \frac{\sqrt{50}}{\sqrt{2}}=\sqrt{\frac{50}{2}}=\sqrt{25}=5. \]

When the radicals have different indices, one may resort to reduction to a common index before multiplying or dividing them.

For instance:

\[ \sqrt{2}\cdot\sqrt[3]{2} = \sqrt[6]{2^3}\cdot\sqrt[6]{2^2} = \sqrt[6]{2^5} = \sqrt[6]{32}. \]

Addition and subtraction

Addition and subtraction of radicals are not carried out by combining the terms under a single radical. One may add or subtract directly only like radicals, that is, radicals with the same index and the same radicand.

In general:

\[ p\sqrt[n]{a}+q\sqrt[n]{a}=(p+q)\sqrt[n]{a}. \]

Likewise:

\[ p\sqrt[n]{a}-q\sqrt[n]{a}=(p-q)\sqrt[n]{a}. \]

For instance:

\[ 3\sqrt{2}+5\sqrt{2}=8\sqrt{2}. \]

Sometimes radicals are not like radicals at the outset, but become so after simplification. For instance:

\[ \sqrt{12}+\sqrt{27} = 2\sqrt{3}+3\sqrt{3} = 5\sqrt{3}. \]

Another example is:

\[ \sqrt{8}-\sqrt{2}+\sqrt{18} = 2\sqrt{2}-\sqrt{2}+3\sqrt{2} = 4\sqrt{2}. \]

By contrast, radicals such as \(\sqrt{2}\) and \(\sqrt{3}\) are not like radicals and cannot be combined into a single radical.

Powers of radicals

Powers of radicals are handled by applying the properties of powers while taking into account the conditions of existence.

If \(\sqrt[n]{a}\) is defined and \(m\in\mathbb{N}^*\), then:

\[ \left(\sqrt[n]{a}\right)^m=\sqrt[n]{a^m}. \]

If moreover \(a\geq0\), we may link this notation to powers with rational exponent:

\[ \sqrt[n]{a^m}=a^{\frac{m}{n}}. \]

Square of a binomial involving radicals

If \(a\geq0\) and \(b\geq0\), then:

\[ (\sqrt{a}+\sqrt{b})^2 = a+2\sqrt{ab}+b. \]

Likewise:

\[ (\sqrt{a}-\sqrt{b})^2 = a-2\sqrt{ab}+b. \]

For instance:

\[ (\sqrt{5}+2)^2 = 5+4\sqrt{5}+4 = 9+4\sqrt{5}. \]

Product of conjugate expressions

If \(a\geq0\) and \(b\geq0\), the product of two conjugate expressions allows one to eliminate square roots:

\[ (\sqrt{a}+\sqrt{b})(\sqrt{a}-\sqrt{b})=a-b. \]

For instance:

\[ (\sqrt{7}+\sqrt{3})(\sqrt{7}-\sqrt{3})=7-3=4. \]

Rationalisation of the denominator

Rationalisation of the denominator consists in transforming a fraction into an equivalent fraction whose denominator contains no radicals.

The underlying idea is to multiply the numerator and denominator by a suitable factor that eliminates the radical from the denominator. The value of the fraction does not change, since one is multiplying by a quantity equal to \(1\).

Denominator with a single square root

Consider a fraction of the form

\[ \frac{c}{\sqrt{a}}, \]

with \(a>0\). To eliminate the root from the denominator, we multiply numerator and denominator by \(\sqrt{a}\):

\[ \frac{c}{\sqrt{a}} = \frac{c\sqrt{a}}{\sqrt{a}\sqrt{a}} = \frac{c\sqrt{a}}{a}. \]

For instance:

\[ \frac{3}{\sqrt{5}} = \frac{3\sqrt{5}}{5}. \]

Denominator with a single \(n\)th root

More generally, if the denominator contains an \(n\)th root, we may use the property of powers.

If the radical is defined and \(\sqrt[n]{a}\neq0\), then:

\[ \frac{c}{\sqrt[n]{a}} = \frac{c\sqrt[n]{a^{n-1}}}{a}. \]

In the case of an even index one must require \(a>0\), whereas in the case of an odd index it suffices to require \(a\neq0\).

For instance:

\[ \frac{1}{\sqrt[3]{2}} = \frac{\sqrt[3]{2^2}}{2} = \frac{\sqrt[3]{4}}{2}. \]

Binomial denominator with square roots

If the denominator is a binomial involving square roots, one uses the product of two conjugate expressions.

For instance, if \(a\geq0\), \(b\geq0\) and \(a\neq b\), then:

\[ \frac{c}{\sqrt{a}+\sqrt{b}} = \frac{c(\sqrt{a}-\sqrt{b})}{(\sqrt{a}+\sqrt{b})(\sqrt{a}-\sqrt{b})} = \frac{c(\sqrt{a}-\sqrt{b})}{a-b}. \]

Likewise:

\[ \frac{c}{\sqrt{a}-\sqrt{b}} = \frac{c(\sqrt{a}+\sqrt{b})}{a-b}. \]

Let us consider an example:

\[ \frac{4}{\sqrt{3}+\sqrt{2}} = \frac{4(\sqrt{3}-\sqrt{2})}{3-2} = 4\sqrt{3}-4\sqrt{2}. \]

Another example is:

\[ \frac{1}{1+\sqrt{5}} = \frac{1-\sqrt{5}}{(1+\sqrt{5})(1-\sqrt{5})} = \frac{1-\sqrt{5}}{1-5} = \frac{\sqrt{5}-1}{4}. \]

Denominator with cube roots

When the denominator contains cube roots, one uses the identities for the difference and sum of cubes:

\[ x^3-y^3=(x-y)(x^2+xy+y^2), \]

and

\[ x^3+y^3=(x+y)(x^2-xy+y^2). \]

For instance, if \(a\neq b\), then:

\[ \frac{1}{\sqrt[3]{a}-\sqrt[3]{b}} = \frac{\sqrt[3]{a^2}+\sqrt[3]{ab}+\sqrt[3]{b^2}}{a-b}. \]

Indeed, setting \(x=\sqrt[3]{a}\) and \(y=\sqrt[3]{b}\), we have

\[ (x-y)(x^2+xy+y^2)=x^3-y^3=a-b. \]

For instance:

\[ \frac{1}{\sqrt[3]{2}-1} = \frac{\sqrt[3]{4}+\sqrt[3]{2}+1}{2-1} = \sqrt[3]{4}+\sqrt[3]{2}+1. \]

Likewise, if \(a\neq -b\), then:

\[ \frac{1}{\sqrt[3]{a}+\sqrt[3]{b}} = \frac{\sqrt[3]{a^2}-\sqrt[3]{ab}+\sqrt[3]{b^2}}{a+b}. \]

Radicals involving variables

When the radicand contains variables, the properties of radicals must be applied together with the conditions of existence. In particular, if the index is even, one must impose that the radicand be non-negative.

For instance, the radical

\[ \sqrt{x-1} \]

is defined over the real numbers if and only if

\[ x-1\geq0, \]

that is, for

\[ x\geq1. \]

Absolute value in simplification

When simplifying radicals with real variables, the absolute value arises naturally in radicals with even index.

Indeed, for every \(x\in\mathbb{R}\),

\[ \sqrt{x^2}=|x|. \]

Likewise:

\[ \sqrt[4]{x^4}=|x|. \]

Furthermore:

\[ \sqrt{x^6}=|x^3|. \]

If instead the index is odd, no absolute value appears. For instance:

\[ \sqrt[3]{x^3}=x. \]

More generally:

\[ \sqrt[2k]{x^{2k}}=|x|,\qquad \sqrt[2k+1]{x^{2k+1}}=x. \]

Domain of expressions with several radicals

If an expression contains several radicals, the domain is obtained by imposing all the conditions of existence simultaneously. In other words, the domain is the intersection of the conditions required by the individual radicals.

Consider, for instance, the function

\[ f(x)=\sqrt{x+2}+\sqrt{4-x}. \]

The first radical requires

\[ x+2\geq0, \]

that is,

\[ x\geq -2. \]

The second radical requires

\[ 4-x\geq0, \]

that is,

\[ x\leq4. \]

Hence the domain is

\[ [-2,4]. \]

Irrational equations

Irrational equations are equations in which the unknown appears under a root sign. To solve them one must pay particular attention to the domain and to any extraneous solutions introduced along the way.

A general method consists of the following steps.

1. Determine the conditions of existence of all the radicals present.
2. Isolate, when possible, one of the radicals.
3. Raise both sides to the appropriate power.
4. Solve the resulting algebraic equation.
5. Check the candidate solutions in the original equation.

This final check is indispensable, since raising both sides to a power can introduce extraneous solutions.

Example with a square root

Consider the equation

\[ \sqrt{2x-1}=x-2. \]

Before squaring, let us determine the necessary conditions. The radical requires

\[ 2x-1\geq0, \]

that is,

\[ x\geq\frac{1}{2}. \]

Moreover, since a square root is always non-negative, the right-hand side must also be non-negative:

\[ x-2\geq0. \]

We must therefore have

\[ x\geq2. \]

Squaring both sides, we obtain

\[ 2x-1=(x-2)^2. \]

Expanding:

\[ 2x-1=x^2-4x+4. \]

Bringing all terms to the right-hand side:

\[ x^2-6x+5=0. \]

Hence

\[ x=1 \quad \text{or} \quad x=5. \]

The value \(x=1\) does not satisfy the condition \(x\geq2\), and is therefore discarded. Let us check \(x=5\) in the original equation:

\[ \sqrt{2\cdot5-1}=5-2. \]

Indeed:

\[ \sqrt{9}=3. \]

Hence the only solution is

\[ x=5. \]

Example with two radicals

Consider now the equation

\[ \sqrt{x+5}-\sqrt{x}=1. \]

The conditions of existence are

\[ x+5\geq0 \qquad\text{and}\qquad x\geq0. \]

Hence the domain is

\[ x\geq0. \]

Let us isolate the first radical:

\[ \sqrt{x+5}=\sqrt{x}+1. \]

Squaring:

\[ x+5=(\sqrt{x}+1)^2. \]

Expanding the right-hand side:

\[ x+5=x+2\sqrt{x}+1. \]

Hence:

\[ 4=2\sqrt{x}. \]

It follows that

\[ \sqrt{x}=2 \]

and therefore

\[ x=4. \]

Let us check this in the original equation:

\[ \sqrt{4+5}-\sqrt{4}=3-2=1. \]

The solution is thus

\[ x=4. \]

To practise with further examples, one may turn to the dedicated collection:

Step-by-Step Practice Problems ➤