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Surds : Definition, Rules Of Surd,Types

rules of surds

Surds are important topic in Maths. We are going to discuss here Surds Definition,Types of Surds, Rules of Surds, Conjugate surds, Comparison of Surds, rationalizing surds

What is Surds in Maths?

  • Surds are the nth roots of the positive rational number whose values cannot be determined accurately or basically it can not be expressed as rational number,whole number. So these are irrational numbers.
  • The nth roots is represented as $ \sqrt[n] { } $
  • $ \sqrt [n] {}$ is known radical sign. It is also called radic
  • $\sqrt [n]{a} $ is also expressed as $a^{1/n}$

Examples

$\sqrt 2$ is a surd
$\sqrt[3] 2$ is a surd
$\sqrt 7$ is a surd
$\sqrt 4$ is not surd as it can expressed as 2
$\sqrt {9/4} $ is not surd as it can expressed as 3/2
$\sqrt {\sqrt 2}$ is also surd as it can be expressed as $\sqrt[4] {2}$
$\sqrt {2 + \sqrt 2}$ is not a surd as per the definition but it is a rational numbers.

Get Surds Questions with detailed answers and solutions

Types Of surds

(A) Classification based on definition

Pure Surds
The surd will only one Rational number is called the Pure Surds
Example
$\sqrt 2$ and $\sqrt 3$ are example of Pure Surds

Mixed Surds
Surds which are expressed as product of Rational number and Irrational numbers.
Example
$2 \sqrt 2$ and $4 \sqrt 3$ are example of Mixed Surds

How to convert Mixed surds into Pure Surds
We can move the rational number inside the surd and make it Pure Surds

Example
$2 \sqrt 2$
Taking the Rational Number inside the Surd
$2 \sqrt 2= \sqrt {2^2 \times 2 } = \sqrt 8$

So , we need to raise the rational number to the same power as that of root
More Examples
(a) $2\sqrt[4] {3} = \sqrt[4] {2^4 \times 3 } =\sqrt[4] { 48}$
(b) $5\sqrt {3} = \sqrt {5^2 \times 3 } =\sqrt { 75}$
(c) $3\sqrt[4] {7} = \sqrt[4] {3^4 \times 4 } =\sqrt[4] { 567}$
(d) $2 \sqrt {5 \sqrt {2}}= \sqrt {2^2 \times 5 \times \sqrt {2}} = 2 \sqrt { \sqrt {20^2 \times 2}} = \sqrt { \sqrt {800} } = \sqrt { 800^{1/2}}= \sqrt [4] {800}$

How to convert Pure surds into Mixed Surds
Many times, the rational number inside the surd can be prime factorized and These prime factors can be taken out provided , the exponent of the prime factor is divisible by root power

Examples
(a) $\sqrt [3] {56} = \sqrt [3]{ 2^3 \times 7} = 2 \sqrt [3] {7}$
(b) $\sqrt [5] {160} = \sqrt [5] {2^5 \times 5} = 2 \sqrt [5] {5}$

(B) Classification Based on number of terms in it

(a) Monomial Surd
A surd having one term only is called Monomial Surd

Example
$2 \sqrt 2$ , $\sqrt[3] 2$

(b) Binomial Surd
A surd having two term only is called Binomial Surd

Example
$2 \sqrt 2 + \sqrt[3] 2$

(c) Trinomial Surd
A surd having three term only is called Trinomial Surd

$\sqrt 3 + \sqrt 2 + \sqrt 5$

Conjugate Surds

Two binomial surds diferring only in the sign between sign is called Conjugate Surds

Example
$\sqrt 3 + \sqrt 2$ and $\sqrt 3 – \sqrt 2$ are Conjugate Surd
$\sqrt 5 + \sqrt 2$ and $\sqrt 5 – \sqrt 2$ are Conjugate Surd

Rules of Surds

1.$(\sqrt [n] {a})^n = a$
Examples
$(\sqrt {2})^2 = 2$
$(\sqrt [3]{3})^3 = 3$

2. $(\sqrt [n] {a})(\sqrt [n] {b}) = \sqrt [n] {ab}$

3. $\frac {\sqrt [n] {a}}{ \sqrt [n] {b}} = \sqrt [n] {\frac {a}{b}}$

4. $a \sqrt c \pm b \sqrt c =(a \pm b) \sqrt c$

5. If a and b are both rationals and $\sqrt {x}$ and $\sqrt y$ are both surds and
$a + \sqrt x = b + \sqrt y$ then a = b and x = y

6. $(\sqrt a + \sqrt b) (\sqrt a – \sqrt b) = a -b$

7. Rationalizing the Surds. We multiply and divide by the conjugate surds of the denominators
$\frac {1}{ \sqrt a + \sqrt b} = \frac {1}{ \sqrt a + \sqrt b} \times \frac {\sqrt a – \sqrt b}{ \sqrt a – \sqrt b} = \frac { \sqrt a – \sqrt b}{ a -b }$

8. $\sqrt [n] {a} = a^{1/n} = a^{m/mn}= \sqrt [mn] {a^m}$

Comparison Of Surds

Surds can be compared only when the power of the surds are same.
Example
$\sqrt 3 < \sqrt 5 < \sqrt 11$

In case the power is different,we need to convert into same power and then perform the comparison
Example
$\sqrt [3] 2$ , $\sqrt [4] {7}$

For converting into same power, we find the LCM of the root power
Surd power of $\sqrt [3] 2$ is 3
Surd power of $\sqrt [4] {7}$ is 4
LCM of 3 and 4 is 12
Now converting surds into surd power of 12
$\sqrt [3] 2 = \sqrt [12] { 2^4} = \sqrt [12] { 16}$
$\sqrt [4] 7 = \sqrt [12] { 7^3} = \sqrt [12] { 343}$
Now surds can be compared
$\sqrt [12] { 343} > \sqrt [12] { 16}$

Rationalizing of Surds

  • When we get a rational number by multiplying two surds, then each of the surds is known as rationalizing factors.
  • So if a surds is given , we can rationalize it by multiplying it another surd.
  • There can be many surds which can rationalize, we will choose the simplest one always

Example
$\sqrt {3}$
Now $\sqrt {3} \times \sqrt {3} =3$
So $\sqrt {3}$ is the simplest rationalizing factor

Solved Examples

1.Rationalize the Denominator
$ \frac {1}{2 – \sqrt 3}$

Solution
$ \frac {1}{2 – \sqrt 3} = \frac {1}{2 – \sqrt 3} \times \frac {2 + \sqrt 3}{2 -\sqrt 3}= \frac {2 + \sqrt 3}{4 -3}= 2 + \sqrt 3 $

2. $\sqrt [3] {32} \times \sqrt [3] {16}$

Solution
$\sqrt [3] {32} \times \sqrt [3] {16} = \sqrt [3] { 32 \times 16} = \sqrt [3] {2^9}= 8$

3. $( \sqrt 3 + 2 \sqrt 2) – ( 3 \sqrt 2 – 2 \sqrt 3)$

Solution
$( \sqrt 3 + 2 \sqrt 2) – ( 3 \sqrt 2 – 2 \sqrt 3)$
$=\sqrt 3 + 2 \sqrt 3 + 2 \sqrt 2 – 3 \sqrt 2$
$=3 \sqrt 3 – \sqrt 2$

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