Question 1134898

One way to look at this is to take two radicals: {{{sqrt(a)}}} and {{{root(3,b)}}}, which differing indices. 

But we can introduce the {{{LCM }}}of {{{2}}} and {{{3}}} which is {{{6}}} and write 
{{{sqrt(a)=root(6,a^3)}}} and {{{root(3,b)=root(6,b^2)}}} where the number {{{2}}} and {{{3}}} is the {{{index}}}.

This introduces a {{{common}}} radical  -  {{{6}}}th root.

 Now we multiply under the {{{same}}}{{{ index}}}: 

{{{sqrt(a)*root(3,b)=root(6,a^3)*root(6,b^2)=root(6,a^3*b^2)}}}.

So in the general case we need the {{{LCM}}} of the indices as the{{{ common}}} radical and then we use the power indices under the common radical. 

So it is {{{not}}}{{{ necessary}}} for {{{2}}} radical expressions to have the {{{same}}}{{{ index}}} in order to {{{multiply}}} them.

NUMERICAL EXAMPLE: 

{{{sqrt(121)}}} and {{{root(3,27)}}}

{{{sqrt(121)=11}}} and {{{root(3,27)=3}}}

 We know the answer is {{{11*3=33}}}. 

But let's see if we get the same result by putting {{{a=121}}} and{{{ b=27}}} in the formula above.

 {{{sqrt(121)*root(3,27)}}}

={{{root(6,(11^2)^3)*root(6,(3^3)^2)}}}

={{{root(6,177561*729)}}

={{{11*3}}}

={{{33}}}

 So the formula works.