Surds are the values in square root that cannot be further simplified into whole numbers( \(\mathbb{W}\) ) or integers( \(\mathbb{Z}\) ).

Surds are irrational numbers as these values cannot be further simplified. On their further simplification, we get decimal values.

Index

**History**

It appears that European mathematician *Gherardo* of *Cremona* (c. 1150) adopted the terminology of surds.

The Greek for “irrational” gets translated into Latin as *surd*, which again is used for irrational numbers.

Fibonacci also adopted the same term in 1202 to refer to a number that has no root.

**Definition of Surds & Indices**

Surds are square roots of numbers that cannot be simplified into a whole number(W) or rational number(𝕫). It cannot accurately be represented in a fraction.

(OR)

A surd is a root of the whole number that has an irrational value.

An index number is a number that is raised to a power.** **The index tells how many times the number multiply by itself.

**Types of Surds**

The different types of surds are as follows:

**Simple Surds**– Surd, which have a single term only, are called a monomial or simple surd.

Example: \(\sqrt{2}, 2\sqrt{2}, …\)

**Pure Surds**– A surd having no rational factor except unity is called a pure surd or complete surd.

Example: \(\sqrt{3}, \sqrt{7}\)

**Similar Surds**– The surd having the same common surd factor are called similar surd or like surd.

Example: \(2–\sqrt{222}, 22–\sqrt{2222}\)

**Mixed Surds**– A surd having a rational co-efficient other than unity is called a mixed surd.

Example: \(2\sqrt{7}, 3\sqrt{6}\)

**Compound Surds**– The algebraic sum of a rational number and simple surd is called a compound surd.

Example: \((\sqrt{5} + \sqrt{7}), (\sqrt[3]{7} + \sqrt[4]{6})\)

**Binomial Surds**– A sum of two roots of rational numbers, at least one of which is an irrational number.

Example: \((4\sqrt{7}+ \sqrt{2}) \mbox{ and } (4\sqrt{7} – \sqrt{2})\)

**Properties of Indices**

**Multiplication Rule**

- \(x^n * x^m = x^{m+n}\)
- \(x^n * y^n = (x * y)^n\)

**Division Rule**

- \(\frac{x^n}{y^n} = (\frac{x}{y})^n\)
- \(\frac{x^n}{x^m} = x^{n-m}\)

**Power Rule**

- \((x^n)^m = x^{m*n}\)
- \(x^{n^{m}} = x^{(n^{m})}\)
- \(\sqrt[m]{x^n} = x^{\frac{n}{m}}\)
- \(\sqrt[m]{x} = x^{\frac{1}{m}}\)
- \(x^{-n} = \frac{1}{x^n}\)

**Applications of Surds & Indices**

- Surds and Indices are used to make sure that important calculations are precise.
- Indices are used in Computer Game, Physics, pH, Accounting, Finance, and many other disciplines.
- Exponential Growth is a critically important aspect of Finance, Demographics, Biology, Economics, Resources, Electronics and many other areas, and indices are regularly used in these and many other fields.

**Examples**

**Question 1.** Rationalise the denominator: \(1/[(3\sqrt11 )- (2\sqrt5)]\).

**Solution.** To rationalize the denominator of the given fraction, multiply the conjugate of denominator on both numerator and denominator.

\frac{1}{(3\sqrt11)-(2\sqrt5)} \\

= \frac{1}{(3\sqrt11)-(2\sqrt5)} \times \frac{(3\sqrt11)+(2\sqrt5)}{(3\sqrt11)+(2\sqrt5)}\\

= \frac{(3\sqrt11)+(2\sqrt5)}{(3\sqrt11)^2-(2\sqrt5)^2} \\

= \frac{(3\sqrt11)+(2\sqrt5)}{99 – 20} \\

= \frac{(3\sqrt11)+(2\sqrt5)}{79} \\

\)

**Question 2.** Divide \(\sqrt{46}\) by \(\sqrt{12}\).

**Solution.**

\frac{\sqrt{46}}{\sqrt{12}}\\

= \sqrt{\frac{46}{12}} \\

= \sqrt{\frac{23}{6}} \\

= \sqrt{\frac{23 \times 6}{6 \times 6}} \\

= \frac{\sqrt{138}}{6} \\

\)

**Question 3.** \(5^{3x-2} = 625\), find \(x\).

**Solution.**

\(5^{3x-2} = 5^4\)

Therefore, we can say that,

\(3x – 2 = 4\)

\(3x = 6\)

\(x = 2\)

**FAQs**

**What is Surd?**Surds are the values in square root that cannot be further simplified into whole numbers( \(\mathbb{W}\) ) or integers( \(\mathbb{Z}\) ).

**Is \(\sqrt{9}\) a surd?**\(\sqrt{9} = 3\),

hence, it is not a surd.

**What are the indices laws?**Indices have three rule multiplication, division, and power rules.

**How do you write Surd in indices form?**If we take an example such as

\(\sqrt[n]{x}\), which is a surd. We can rewrite it in indices form as

\(\sqrt[n]{x} = x^{\frac{1}{n}}\), which is Indice form.