Surds and Indices

Surds and indices chapter plays a crucial role in simplification problems of arithmetic. Here we would learn about its basic concepts and tricks to handle these problems efficiently in competitive examinations.

Surds: A surd is a root of a rational number with an irrational value. In other words, when it is a root and irrational, it is surd. Surds can not be further simplified into whole numbers or integers.

Example: \sqrt{2}, \sqrt{3}, \sqrt[3]{5}, \sqrt{\frac{1}{7}}, etc.

Note: An irrational number is a real number that can not be written as a simple fraction.

Example: \pi, \sqrt{2},\sqrt{3}, etc.

If N is an irrational number then N\neq\frac{p}{q}, where p and q are integers and q\neq0.

Classification of Numbers
Classification of Numbers

Types of Surds

  • Pure Surds: When surds have only one irrational number. 

            Example: \sqrt{2}, \sqrt{11}, \sqrt{17}, etc.

  • Mixed Surds: When surds have both rational and irrational numbers. 

            Example: 2\sqrt{5}, 4\sqrt{7}, 8\sqrt{3}, etc.

  • Compound Surds: When there are two or more surds in one mathematical expression.

            Example: 4+\sqrt{5}, \sqrt{5}+\sqrt{6}, 3+2\sqrt{6}, etc.

Indices: The index (Plural: Indices) is the power (exponent) of a number.

Example: 2^{5} has an index of 5 and 2 is its base. Here exponent 5 defines, how many times 2 has been multiplied by itself.

Surds and Indices
Rational and Irrational Numbers

Difference Between Rational and Irrational Numbers

Rational NumberIrrational Number
Rational Numbers can be expressed in the form of \frac{p}{q}, where p and q are integers and q\neq0.Irrational Numbers can’t be expressed in \frac{p}{q} form.
Rational Numbers include finite and recurring decimals.Irrational Numbers consist of non-recurring and non-terminating decimals.
Square roots of perfect squares and cube roots of perfect cubes are rational.It includes surds.
Examples: -3 ,0 ,1 ,\frac{5}{8}, \sqrt{25}, \sqrt{\frac{1}{9}}, etc.Examples: \pi, e ,\sqrt{7} ,\sqrt{\frac{2}{5}} ,etc.
Difference Between Rational and Irrational Numbers

Rules of Surds and Indices

If a and b are any two real numbers, m and n are integers then

(i) a^{m}\times a^{n}=a^{m+n}

    Example: 2^{3}\times 2^{5}=2^{8}=256

(ii) \frac{a^{m}}{a^{n}}=a^{m-n}

    Example: \frac{5^{3}}{5^{4}}=5^{-1}=\frac{1}{5}

(iii) (ab)^{m}=a^{m}\times b^{m}

(iv) x^{m^{n}}=x^{(m^{n})}

(v) (a^{m})^{n}=a^{m\times n}

(vi) a^{-m}=\frac{1}{a^{m}}

(vii) a^{0}=1

(viii) \sqrt[n]{a}=a^{\frac{1}{n}}

(ix) \sqrt[n]{x^m}=x^{\frac{m}{n}}

(x) If x^{\frac{1}{n}}=a then x=a^{n}

(xi) \sqrt{a}\displaystyle\pm \sqrt{b} \neq \sqrt{a\displaystyle\pm b}

(xii) \sqrt{a}\times \sqrt{b}=\sqrt{ab}

(xiii) \frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}

Rationalizing Surds

Rationalization is the process of changing the denominator of a fraction to a rational number.

Que: Rationalize \frac{1}{\sqrt{5}+\sqrt{3}}.

Solution: Multiply the conjugate of the denominator by both, the numerator and the denominator.

=\frac{1\times (\sqrt{5}-\sqrt{3})}{(\sqrt{5}+\sqrt{3})\times (\sqrt{5}-\sqrt{3})}


=\frac{1}{2}\times (\sqrt{5}-\sqrt{3})

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Surds and Indices problems

Que 1: Find the value of \frac{(243)^\frac{n}{5}\times 3^{2n+1}}{9^{n}\times 3^{n-1}}.

Solution: Numerator =(3^{5})^{\frac{n}{5}}\times 3^{2n+1}=3^{n}\times 3^{2n+1}=3^{3n+1}

Denominator=3^{2n}\times 3^{n-1}=3^{3n-1}


Que 2: If x^{x\sqrt{x}}=(x\sqrt{x})^{x} then find the value of x?

Solution: x^{x^{1}\times x^{\frac{1}{2}}}=(x^{1}\times x^{\frac{1}{2}})^{x}.


So, x^{\frac{3}{2}}=\frac{3x}{2}.

After squaring both sides of the equation,



Now, x=\frac{9}{4} is the correct answer (x=0 would give an indeterminate value)

Que 3:  Find the value of \frac{5^{n+3}-6\times 5^{n+1}}{9\times 5^{n}-5^{n}\times 2^{2}}?

Solution: \frac{5^{n}(5^{3}-6\times 5)}{5^{n}(9-4)}.


Que 4: Find the value of [(\sqrt[3]{256^{2}})^{\frac{3}{2}}]^{\frac{1}{4}}?

Solution: 256=2^{8}

=[(2^{\frac{8\times 2}{3}})^{\frac{3}{2}}]^{\frac{1}{4}}


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Que 5:  Find the value of \sqrt[3]{4\frac{12}{125}}?

Solution: 4\frac{12}{125}=\frac{512}{125}=(\frac{8}{5})^{3}

 So, The correct answer=\frac{8}{5}=1\frac{3}{5}

Que 6: If \frac{\sqrt{4356\times \sqrt{x}}}{\sqrt{6084}}=11, then find the value of x ?

Solution: After squaring both sides of the equation, 




Note: Click here for Quick Calculation Tips and Tricks.

Que 7: Arrange the following in ascending order if a>1

(i) \sqrt[3]{\sqrt[4]{a^{3}}}

(ii) \sqrt[3]{\sqrt[5]{a^{2}}}

(iii) \sqrt{\sqrt[3]{a}}

(iv) \sqrt{\sqrt[5]{a^{3}}}

Solution: \sqrt[3]{\sqrt[4]{a^{3}}}=(\sqrt[4]{a^{3}})^{\frac{1}{3}}=(a^{\frac{3}{4}})^{\frac{1}{3}}=a^{\frac{1}{4}}.




Now compare exponents,

(\frac{1}{4}, \frac{4}{15}, \frac{1}{6}, \frac{3}{10})\times 60=(15, 16, 10, 18).           

Here, LCM(4,15,6,10)=60

Now, 10<15<16<18

So, \frac{1}{6}< \frac{1}{4}<\frac{4}{15}<\frac{3}{10} 

 Correct ascending order= (iii)<(i)<(ii)<(iv)

Que 8: Find the square root of \sqrt{5+2\sqrt{6}} ?

Solution: \sqrt{(\sqrt{3})^{2}+(\sqrt{2})^{2}+2\times\sqrt{3}\times\sqrt{2}}.



Important results of Surds and Indices

1. If N=\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+......}}}}

    Then N=\frac{1+\sqrt{1+4x}}{2}

Proof: N=\sqrt{x+N}.



N=\frac{1\displaystyle\pm\sqrt{1+4x}}{2}.   (Sridharacharya Formula)

Note: N has a positive value so take the positive sign here. 

2. If N=\sqrt{x-\sqrt{x-\sqrt{x-\sqrt{x-......}}}}

  Then N=\frac{-1+\sqrt{1+4x}}{2}

3. If N=\sqrt{x\displaystyle\pm\sqrt{x\displaystyle\pm\sqrt{x\displaystyle\pm\sqrt{x\displaystyle\pm......}}}} then find the value of N?

Here x=k×(k+1), k is a Natural Number (Positive Integer)

Case 1: (+) sign


Case 2: (-) sign


 Example: N=\sqrt{12+\sqrt{12+\sqrt{12+\sqrt{12+......}}}}=4


Here, 12=3×4

4. If N=\sqrt{x\sqrt{x\sqrt{x\sqrt{x......}}}} then N=x

Proof: N=\sqrt{x.N}.



So, N=x    (For x>0, N can’t be 0)

5. If K=\sqrt{x\sqrt{x\sqrt{x\sqrt{x.....n_{th}term}}}} then find the value of K ?

Number of terms= Number of square roots=n


Example: K=\sqrt{5\sqrt{5\sqrt{5\sqrt{5}}}}=(5)^{\frac{2^{4}-1}{2^{4}}}=(5)^{\frac{15}{16}}.

Question: \frac{\sqrt{100!\sqrt{100!\sqrt{100!\sqrt{100!}}}}}{\sqrt{99!\sqrt{99!\sqrt{99!\sqrt{99!}}}}}=\sqrt[16]{10^x}

Find the value of x?

Solution: Numerator=(100!)^\frac{15}{16}.



The value of x=30

Frequently Asked Questions

  1. What is the difference between Surds and Indices?

    Surds can be expressed as the nth root of a rational number while Indices refer to the power to which a number is raised.

  2. Is surd an irrational number?

    Yes, Surd is irrational because, in decimal form, it goes on without repetition.

  3. How can we identify Surds?

    A rational number that contains a radical sign (\sqrt[n]{}) and its solution is an irrational value, called Surd.

  4. Can a Surd be expressed in Index form?

    Yes, A surd can be expressed in the form of a fractional index.
    Surd Form: \sqrt[n]{a}
    Index Form: a^{\frac{1}{n}}

  5. The square root of ‘pi’ is a Surd or not?

    No, \pi is an irrational number but a Surd is the nth root of a rational number.

Classification of Numbers: Click Here

Multiple Choice Questions (MCQ)

35 Questions with solutions, based on the concepts of Surds and Indices.


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