The surds play an important role to study many areas of Mathematics; for example, solving algebraic equations, simplifying radical expressions, etc. In this article, we will learn the properties of surds or radicals.

Table of Contents

## Radicals as Exponents

A radical of index n is written with the exponent 1/n. For example,

Index | Expression as a Radical |

n | $\sqrt[n]{x}=x^{1/n}$ |

2 | √x = x^{1/2} |

3 | $\sqrt[3]{x}=x^{1/3}$ |

In the definition of radicals of index n, that is, $\sqrt[n]{x}=x^{1/n}$, the number x is called the radicand. Thus, radicand is the number that lies inside the radical.

## Equality of Radicals

If two quadratic radicals (i.e., index 2) √x and √y are equal, then we must have x=y if either positive or negative values are considered together, i.e., √x=√y ⇔ x=y.

For radicals of higher index ≥ 3, we also have $\sqrt[n]{x}=\sqrt[n]{y}$ ⇔ x=y.

## Addition & Subtraction of Radicals

We can add (or subtract) two or more radicals if they are similar radicals having similar radicands. For example, we can add 3√2 and 4√2.

3√2+4√2 = (3+4)√2 = 7√2.

Likewise, 3√2-4√2 = (3-4)√2 = -√2.

But, 3√2 and 5√3 cannot be added in the above manner.

## Multiplication of Radicals

The multiplication of two radicals of the same index n is also a radical of the index n. This multiplication rule of radicals is given as follows:

$\sqrt[n]{x} \times \sqrt[n]{y}=\sqrt[n]{xy}$.

For radicals of index 2, that is, for square roots the multiplication rule is given below:

√(a×b) = √a × √b.

For example, √10 = √(2×5) = √2 × √5. Note that if a=b then we have:

√(a×a) = a.

This shows that the square root of a^{2} is a. For example, √16 = √(4×4) = 4.

**Also Read:**

**Surds: Definition, Rules, Types, Examples**

## Division of Radicals

The division of two radicals of the same index n is also a radical of the index n, and the rule is given below:

$\sqrt[n]{\dfrac{x}{y}}=\dfrac{\sqrt[n]{x}}{\sqrt[n]{y}}$.

For radicals of index 2, that is, for square roots we have:

$\sqrt{\dfrac{x}{y}}=\dfrac{\sqrt{x}}{\sqrt{y}}$.

For example, $\sqrt{\dfrac{32}{2}}=\dfrac{\sqrt{32}}{\sqrt{2}}$ $=\dfrac{4\sqrt{2}}{\sqrt{2}}=4$. Note that $\sqrt{\dfrac{32}{2}}=\sqrt{16}=4$.

## Negative Radicand with Index 2

If we have a negative radicand with index 2, then it is the square root of a negative number. This is considered an imaginary number.

For example, $\sqrt{-1}$ is a purely imaginary number.

## Square Root of a Perfect Square

The square root of a perfect square (a number that is a square of another number) is always a natural number.

For example, 16 is a perfect square, and √16=√(4×4)=4.

## Perfect Radical

Consider a radical of index n and the radicand is the product of n copies of a number b, then that radical is called a perfect radical and it is equal to b. Here,

$\sqrt[n]{b^n}=b$.

For example, $\sqrt[3]{3^3}=\sqrt{3 \times 3 \times 3}$ $=3$. So we can say that 3^{3} = 27 is a perfect cube number as the index is 3 here.

## Radical Expression

An expression containing at least one radical is called a radical expression. An expression involving radicals of index n only can be made in its simplest form, called simplified radical form, if the following conditions are satisfied:

- No radicands can be fractions.
- If the radical expression is a fraction, then it should not contain radicals as a denominator.
- The factors of the radicands cannot be perfect nth powers.

For example, √2+3√5 is a radical expression.

**Also Read:**

## Moving Exponent n to Other Side

If we move an exponent n from one side of an equation to the other side, then we will get a radical of index 1/n.

For example, x^{n}=a ⇒ x=a^{1/n}.

## Moving Exponent 1/n to Other Side

Similarly, if we move an exponent 1/n (that is, a radical of index n) from one side of an equation to the other side, then we will get an exponent n.

For example, x^{1/n}=a ⇒ x=a^{n}.

## Solved Examples on Radicals

**Question 1:** Multiply root 2 with root 3.

*Solution: *

√2 × √3 =√(2×3) = √6.

## FAQs on Radicals

**Q1: What are radicals in Mathematics?**

Answer: The radicals with index n in mathematics is usually referred as a nth root. Thus, square roots are examples of radicals of index 2.

**Q2: Give some examples of radicals.**

Answer: √2, √3 are daricals of inedx 2.

This article is written by Dr. T. Mandal, Ph.D in Mathematics. On Mathstoon.com you will find Maths from very basic level to advanced level. Thanks for visiting.