Properties of Radicals | Radicals Properties

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.

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 = x1/2
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 a² is a. For example, √16 = √(4×4) = 4.

Also Read:

Surds: Definition, Rules, Types, Examples

Order of Surds

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 3³ = 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:

Like and Unlike Surds

Pure and Mixed Surds

Simple and Compound Surds

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, xⁿ=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=aⁿ.

Solved Examples on Radicals

Question 1: Multiply root 2 with root 3.

Solution:

√2 × √3 =√(2×3) = √6.

FAQs on Radicals