Nine Standard >> Operations on surds | Part-2

Operation on surds:

(sqrt{a}+sqrt{b}neq sqrt{ab})
(sqrt{2}+sqrt{3}neq sqrt{5})

(sqrt{a}.sqrt{b}= sqrt{ab})
(sqrt{2}.sqrt{3}= sqrt{6})

Surds of the same order can be multiplied

(sqrt[n]{x}.sqrt[n]{y}=x^{frac{1}{n}}.y^{frac{1}{n}}=(xy)^{frac{1}{n}}=sqrt[n]{xy})
(sqrt[n]{x}.sqrt[m]{y}=x^{frac{1}{n}}.y^{frac{1}{m}}=x^{frac{m}{nm}}.y^{frac{n}{nm}})
      (=(x^{m})^{frac{1}{nm}}.(y^{n})^{frac{1}{nm}})=((x^{m}.y^{n})^{frac{1}{nm}})
        =(sqrt[mn]{x^{m}.y^{n}})

Example:

(sqrt[3]{2} times sqrt{3}=2^{frac{1}{3}}.3^{frac{1}{2}})
              =(2^{frac{1 times 2}{3 times 2}}.3^{frac{1 times 3}{2 times 3}}) [ LCM of 2 &3 is 6]
              =(2^{frac{2}{6}}.3^{frac{3}{6}})
              =(sqrt[6]{2^{2}}.sqrt[6]{3^{3}})
              =(sqrt[6]{4 times 27})
              =(sqrt[6]{108})