**What Is Surd Or Radical**

**Real Number Line:**

Irrational numbers like √2, √3, √5 etc. can be represented by points on the number line. Since all rational numbers and irrational numbers can be represented on the number line, we call the number line as real number line.

**Surds:**

√2, √3, √5, √21, ……………. are irrational numbers, These are square roots (second roots), of some rational numbers, which can not be written as squares of any rational number.

- If a is rational number and n is a positive integer such that the n
^{th}root of a is an irrational number, then a^{1/n}is called a**surd**or**radical**.

**Example:**√2, √3, √5 etc. - If is a surd then ‘n’ is known as order of surd and ‘a’ is known as
**radicand**. - Every surd is an irrational number but every irrational number is not a surd.

**Quadratic Surd:**

A surd of order 2 is called a quadratic surd.

**Example:** √3 = 3^{1/2} is a quadratic surd but √9= 9^{1/2} is not a quadratic surd, because √9= 9^{1/2} = 3 is a rational number. So, √9 is not a surd.

**Cubic Surd:**

A surd of order 3 is called a cubic surd.

**Example:** The real number \(\sqrt[3]{4}\) is a cubic surd but the real number \(\sqrt[3]{8}\) is not a cubic surd as it not a surd.

**Biquadratic Surd:**

A surd of order 4 is called a biquadratic surd. A biquadratic surd is also called a quadratic surd.

**Example:** \(\sqrt[4]{5}\) is a biquadratic surd but \(\sqrt[4]{81}\) is not a biquadratic surd as it is not a surd.

**Laws of Radicals:**

- For any positive integer ‘n’ and a positive rational number ‘a’.

- A surd which has unity only as rational factor is called a pure surd.
- A surd which has a rational factor other than unity is called a mixed surd.
- Surds having same irrational factors are called similar or like surds.
- Only similar surds can be added or subtracted by adding or subtracting their rational parts.
- Surds of same order can be multiplied or divided.
- If the surds to be multiplied or to be divided are not of the same order, we first reduce them to the same order and then multiply or divide.
- If the product of two surds is a rational number, then each one of them is called the rationalising factor of the other.
- A surd consisting of one term only is called a
**monomial surd**. - An expression consisting of the sum or difference of two monomial surds or the sum or difference of a monomial surds and a rational number is called
**binomial surd**.

**Example:**\(\sqrt{2}+\sqrt{5},\,\sqrt{3}+2,\,\,\sqrt{2}-\sqrt{3}\) etc. are binomial surds. - The binomial surds which differ only in sign

(+ or –) between the terms connecting them, are called**conjugate surds.****binomial surd**.

**Example:**\(\sqrt{3}+\sqrt{2}\) and \(\sqrt{3}-\sqrt{2}\) or \(2+\sqrt{5}\) and \(2-\sqrt{5}\) are conjugate surds.

**Surd Or Radical Example Problems With Solutions**

**Example 1:** State with reasons which of the following are surds and which are not

(i) √64 (ii) √45 (iii) √20 × √45

\((\text{iv})\text{ }8\sqrt{10}\div 4\sqrt{15}\text{ (v) }3\sqrt{12}\div 6\sqrt{27}\text{ (vi) }\sqrt[3]{5}\times \sqrt[3]{25}\)

**Solution: **(i) √64 = 8

8 is a rational number, hence √64 is not a surd.

(ii) \(\sqrt{45}=\sqrt{9\times 5}=3\sqrt{5}\)

Because the rational number 45 is not the square of any rational number, hence √45 is a surd.

Which is an irrational number.

Because the rational number 8/3 is not the square of any rational number, hence the given expression is a surd.

**Example 2: **Simplify the following

\(\text{(i) }{{\left( \sqrt[3]{5} \right)}^{3}}\text{ (ii) }\sqrt[3]{64}\)

**Solution:**

**Example 3: **Find the value of x in each of the following:

\(\text{(i) }\sqrt[3]{4x-7}-5=0\text{ (ii) }\sqrt[4]{3x+1}=2\)

**Solution:**

**Example 4: **Simplify each of the following:

\(\text{(i) }\sqrt[3]{3}\times \sqrt[3]{4}\text{ (ii) }\sqrt[3]{128}\)

**Solution: **

**Example 5: ** Simplify each of the following:

\(\text{(i) }\sqrt[3]{\frac{8}{27}}\text{ (ii) }\frac{\sqrt[4]{3888}}{\sqrt[4]{48}}\)

**Solution:**

**Example 6: **Siplify each of the following

\(\text{(i) }\sqrt[4]{\sqrt[3]{3}}\text{ (ii) }\sqrt[2]{\sqrt[3]{5}}\)

**Solution: **

**Pure And Mixed Surds:**

**(i) Pure Surd:**

A surd which has unity only as rational factor, the other factor being irrational, is called a pure surd.

**Example:** \( \sqrt{3},\,\,\sqrt[5]{2},\,\,\sqrt[4]{3} \) are pure surds.

**Example:** \( \sqrt[{}]{6},\,\,\sqrt[3]{12} \) are pure surds.

**(ii) Mixed Surd:**

A surd which has a rational factor other than unity, the other factor being irrational, is called a mixed surd.

**Example:** \( 2\sqrt{3},\,\,5\,\sqrt[3]{12},\,\,2\,\sqrt[4]{5} \) are mixed surds.

**TypeI: On expressing of mixed surds into pure surds**

**Example 7: **Express each of the following as a pure surd.

\( \text{(i) 2}\sqrt{3}\text{ (ii) 2}\text{.}\sqrt[3]{4}\)

\( \text{(iii) }\frac{3}{4}\sqrt{32}\text{ (iv) }\frac{3}{4}\sqrt{8} \)

**Solution:**

**Example 8: **Expressed each of the following as pure surds

\(\text{(i) }\frac{2}{3}\sqrt[3]{108}\text{ (ii) }\frac{3}{2}\sqrt[4]{\frac{32}{243}}\)

**Solution:**

**Example 9: **Express each of the following as pure surd

\(\text{(i) a}\sqrt{a+b}\text{ (ii) }a\sqrt[3]{{{b}^{2}}}\text{ (iii)2ab}\sqrt[3]{ab}\)

**Solution:**

**TypeII: On expressing given surds as mixed surds in the simplest form.**

**Example 10: **Express each of the following as mixed surd in its simplest form:

\( \text{(i) }\sqrt{80}\text{ (ii) }\sqrt[3]{72}\text{ (iii) }\sqrt[5]{288} \)

\(\text{(iv) }\sqrt{1350}\text{ (v) }\sqrt[5]{320}\text{ (vi) 5}\text{.}\sqrt[3]{135} \)

**Solution:**

**Example 11: **Express \(\sqrt[4]{1280}\) as mixed surd in its simplest form

**Solution:**