Surds & Indices

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.


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.

Leonardo Pisano Bigollo used surds in 1202.
Leonardo Pisano Bigollo (Source)

Definition of Surds & Indices

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


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:

  1. Simple Surds – Surd, which have a single term only, are called a monomial or simple surd.
    Example: \(\sqrt{2}, 2\sqrt{2}, …\)
  1. Pure Surds – A surd having no rational factor except unity is called a pure surd or complete surd.
    Example: \(\sqrt{3}, \sqrt{7}\)
  1. Similar Surds – The surd having the same common surd factor are called similar surd or like surd.
    Example: \(2–\sqrt{222}, 22–\sqrt{2222}\)
  1. Mixed Surds – A surd having a rational co-efficient other than unity is called a mixed surd.
    Example: \(2\sqrt{7}, 3\sqrt{6}\)
  1. Compound Surds – The algebraic sum of a rational number and simple surd is called a compound surd.
    Example: \((\sqrt{5} + \sqrt{7}), (\sqrt[3]{7} + \sqrt[4]{6})\)
  1. 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

  1. \(x^n * x^m = x^{m+n}\)
  2. \(x^n * y^n = (x * y)^n\)

Division Rule

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

Power Rule

  1. \((x^n)^m = x^{m*n}\)
  2. \(x^{n^{m}} = x^{(n^{m})}\)
  3. \(\sqrt[m]{x^n} = x^{\frac{n}{m}}\)
  4. \(\sqrt[m]{x} = x^{\frac{1}{m}}\)
  5. \(x^{-n} = \frac{1}{x^n}\)

Applications of Surds & Indices

  1. Surds and Indices are used to make sure that important calculations are precise.
  2. Indices are used in Computer Game, Physics, pH, Accounting, Finance, and many other disciplines. 
  3. 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.


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 \(\sqrt{12}\).


= \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 \(x\).


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

Therefore, we can say that,

\(3x – 2 = 4\)
\(3x = 6\)
\(x = 2\)


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.

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