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Properties of Irrationals and Surds

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 ▪ The set  P  of  all irrationals is not closed for addition, since the sum of  two irrationals need not be irrational.

e.g.  ( 5 +\sqrt{3} ) \in P,  ( 5 –\sqrt{3} ) \in P  but  ( 5 + \sqrt{3} ) + ( 5 –\sqrt{3} ) = 10 \notin P.

 ▪ The set P of  all irrational is not closed for multiplication, since the product of  two irrationals need not be irrational.

e.g. \sqrt{2}  \in P,  – \sqrt{2} \in P  but  \sqrt{2} \times (– \sqrt{2})  =  – 2 \notin P.

Surd  :  If  ‘a’  is a rational, ‘n’  is a positive integer and  \sqrt[n]{a} = a^{\frac{1}{n}}  is irrational, then \sqrt[n]{a}  is called  ‘a’ surd of  order ‘n’.

e.g. \sqrt{5}  is a surd of  order  2  &  \sqrt[3]{4}  is a surd of  order  3.

■ Comparison of  Surds  :   If  two surds are of  the same order, then the one whose radicand is larger, is the larger of  the two.

For Example :  \sqrt[3]{26}  > \sqrt[3]{23}

 e.g. Which is larger\sqrt{2}   or \sqrt[3]{3}.

 Soln. Given surds are of  order 2 and 3 respectively whose L.C.M. is 6.

Taking 6th power of  each surd, as shown below.

\left (\sqrt{2} \right )^{6}= \left (2^{\frac{1}{2}} \right )^{6} = 2^{3}  =  8

\left (\sqrt[3]{3} \right )^{6} = \left (3^{\frac{1}{3}} \right )^{6} = 3^{2}  =  9

Clearly,  9  >  8.    So,   > \sqrt[3]{3}  > \sqrt{2}.

 e.g. Simplify : \sqrt[4]{256}   + \sqrt[3]{729} + \sqrt[5]{32}

 Soln.   \sqrt[4]{256}   + \sqrt[3]{729} + \sqrt[5]{32}

=  \sqrt[4]{4\times 4\times4\times4}   + \sqrt[3]{9\times9\times9} + \sqrt[5]{2\times2\times2\times2\times2}

=  (4 + 9 + 2)

=  15.

 

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