Complete detail about Surd

SURD

If the root of a number cannot be exactly obtained, the root is called a surd or an irrational number.

Thus, \sqrt { 2 } ,\sqrt { 3 } ,\sqrt { 5 } ,\sqrt { 7 }, … are all surd or irrational numbers.

            \sqrt [ 3 ]{ 2 } ,\sqrt [ 3 ]{ 3 } ,\sqrt [ 3 ]{ 5 } ,\sqrt [ 3 ]{ 7 }, … are all surd or irrational numbers.

Simply, the positive nth root of number a  is called a surd or  radica;.

For example:

(i) \sqrt { 45 } =\sqrt { 9\times 5 } =3\sqrt { 5 } is an irrational number. Hence, it is a surd.

(ii) \sqrt { 144 }= 12 is a rational number. Hence, it is not a surd.

(iii) \sqrt [ 3 ]{ 5 } \times \sqrt [ 3 ]{ 25 } =\sqrt [ 3 ]{ 5\times 25 } =\sqrt [ 3 ]{ 5\times 5\times 5 } =5 is a rational number.

      Hence, it is not a surd.

(iv) 8\sqrt { 10 } \div 4\sqrt { 15 } =\frac { 8\sqrt { 10 }  }{ 4\sqrt { 15 }  } =\frac { 8 }{ 4 } \sqrt { \frac { 10 }{ 15 } \times \frac { 15 }{ 15 }  } =\frac { 8\sqrt { 150 }  }{ 4\times 15 } .

=\frac { 2 }{ 15 } \sqrt { 5\times 5\times 6 } =\frac { 2\times 5 }{ 15 } \sqrt { 6 } =\frac { 2 }{ 3 } \sqrt { 6 } .

Which is irrational. Hence, it is a surd.

Pure Surd

A surd in which the whole of the rational number is under the radical sign and makes the radicand, is called pure sured.

For example: \sqrt { 8 } ,\sqrt { 5 } ,\sqrt [ 3 ]{ 15 } ,\sqrt [ 3 ]{ 11 } ,\sqrt { 80 } etc.

Mixed Surd

If some part of the quantity under the radical sign is taken out of it, then it make the surd mixed.

For example: \sqrt { 45 } =\sqrt { 9\times 5 } =3\sqrt { 5 }.

Rationalisation of Surd

When the denominator of an expression is a surd which can be reduced to an expression with rational denominator, this process is known as rationalising the denominator of the surd.

When the product of two surds is a rational number, then each of the two surds is called rationalising factor of the other. For example, the rationalising factor of \sqrt { 3 }  is \sqrt { 3 }  and rationalising factor of \sqrt [ 3 ]{ 2 } is\sqrt [ 3 ]{ { 2 }^{ 2 } } or\sqrt [ 3 ]{ 4 }.

Also rationalizing factor of \sqrt [ 3 ]{ { ab }^{ 2 }{ c }^{ 2 } } is\sqrt [ 3 ]{ { a }^{ 2 }bc } because \sqrt [ 3 ]{ { ab }^{ 2 }{ c }^{ 2 } } \times \sqrt [ 3 ]{ { a }^{ 2 }bc } = abc.

Final Wordings:

So Guys, That is it and we are to the end of our complete detail about Surd. And hope you like it. If yes, then share it with your friends using the sharing tools at your end. If you have questions and comments ask us by using the form at the bottom, or send us a mail.

Source-: Ts aggarwal & R.L Arora References


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