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Surd

  • May 30, 2020
  • Category: Surd
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Surds  : We know that \sqrt{2} ,\sqrt{3} , \sqrt{5}  are all irrational numbers, whereas  \sqrt{4} ,\sqrt{9} ,\sqrt{16}  are all rational numbers.

The square root of  ‘a’ is denoted by  \sqrt{a}= a^{\frac{1}{2}}  ;  the cube root, fourth root and fifth roots of  ‘a’ are denoted by  \sqrt[3]{a}  = a^{\frac{1}{3}} ,  \sqrt[4]{a} = a^{\frac{1}{4}} ,  \sqrt[5]{a} =  a^{\frac{1}{5}}  respectively.

Similarly,  \sqrt[n]{a} = a^{\frac{1}{n}}   denotes   nth   root of   ‘a’.

‘a’ is called radicand and  \sqrt[n]{}  is called radical sign.

A  surd is defined  as  any  root  of   a  rational number  which is not an integer.

■ For    to be a surd,

 ▪ If  n  is  even,  then  the radicand  ‘a’ should  be  a  positive rational number.

 ▪ ‘n’  should be a natural number.

 ▪   is an irrational number.

■ \sqrt[n]{a} = a^{\frac{1}{n}}

e.g.,  \sqrt[3]{5} =  5^{\frac{1}{3}}  

 

■ \sqrt[n]{ab} = \sqrt[n]{a} \times \sqrt[n]{b}   

e.g., \sqrt[3]{11\times 5}  =  \sqrt[3]{11} \sqrt[3]{5}

 

■ \sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}

e.g., \sqrt[5]{\frac{4}{5}} = \frac{\sqrt[5]{4}}{\sqrt[5]{5}}

 

■ \left (\sqrt[n]{a} \right )^{n} =  a,   also   \left (\sqrt[n]{a}\sqrt[n]{b} \right )^{n}  =  ab

e.g.,  \left (\sqrt[2]{7} \right )^{2}  =  7.

 

■ \sqrt[m]{\sqrt[n]{a}} = \sqrt[mn]{a}

e.g.,  \sqrt[3]{\sqrt[2]{10}}  =  \sqrt[6]{10}

 

■   \left (\sqrt[n]{a} \right )^{m}= \sqrt[n]{a^{m}}

e.g.,  \left (\sqrt[3]{6} \right )^{2}  =  \sqrt[3]{6^{2}}  =  \sqrt[3]{36}

 

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