## 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:

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Source-: Ts aggarwal & R.L Arora References

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