Surds are numbers left in square root or cube root form, which cannot be reduced to a whole number as the number within the root sign is not a perfect square or a cube and hence it cannot be reduced to a rational number. Obviously surds are irrational numbers and their value is an infinite series of numbers after decimal point.
In fractions they create problems when they are denominator, as dividing by such a number is often cumbersome. If we have just one surd in denominator say #sqrt7#, multiplying numerator and denominator by same surd, here #sqrt7# removes surd in denominator and makes it easy for us to calculate value of fraction. If denominator is say #root(3)3# or #root(3)(3^2)=root(3)9#, we can remove surd by multiplying numerator and denominator by #root(3)9# or #root(3)3# respectively.
But when we have in denominator sum of a rational and irrational number of type #a+sqrtb#, to remove surd, we multiply numerator and denominator by #a-sqrtb#, which is called its conjugate. This process is called rationalization. To rationalize #a-sqrtb#, we multiply numerator and denominator by #a+sqrtb#. Observe that this rationalises using the identity #(a+b)(a-b)=a^2-b^2#
Hence #(5sqrt2)/(3sqrt6+7)=((5sqrt2)(-3sqrt6+7))/((3sqrt6+7)(-3sqrt6+7))#
= #(-15sqrt12+35sqrt2)/(7^2-(3sqrt6)^2)#
= #(-15xx2sqrt3+35sqrt2)/(49-54)#
= #(-30sqrt3+35sqrt2)/-5#
= #6sqrt3-7sqrt2#
If it is #(5sqrt2)/(3sqrt6-7)=((5sqrt2)(3sqrt6+7))/((3sqrt6-7)(3sqrt6+7))#
= #(15sqrt12+35sqrt2)/((3sqrt6)^2-7^2)#
= #(15xx2sqrt3+35sqrt2)/(54-49)#
= #(30sqrt3+35sqrt2)/5#
= #6sqrt3+7sqrt2#
