# Product of two unlike Quadratic Surds

The product of two unlike quadratic surds cannot be rational.

Suppose, let √p and √q be two unlike quadratic surds.

We have to show that √p ∙ √q cannot be rational.

If possible, let us assume, √p ∙ √q = r where r is rational.

Therefore, √q = r/√p = (r ∙ √p)/(√p ∙ √p) =  (r /p) √p

√q = (a rational quantity) √p, [Since, r and p both are rational, therefore, r/p is rational.)

Now from the above expression we clearly see that √p and √q are like surds, which is a contradiction. Therefore, our assumption cannot hold i.e., √p ∙ √q cannot be rational.

Therefore, the product of two unlike quadratic surds cannot be rational.

Notes:

1. In like manner we can show that the quotient of two unlike quadratic surds cannot be rational.

2. The product of two like quadratic surds always represent a rational quantity.

For example, consider two like quadratic surds m√z and n√z where m and n are rational.

Now the product of m√z and n√z = m√z ∙ n√z = mn(√z^2)= mnz, which is a rational quantity.

3. The quotient of two like quadratic surds always represent a rational quantity. For example, consider For example, consider two like quadratic surds m√z and n√z where m and n are rational.

Now the quotient of m√z and n√z = (m√z)/(n√z) = m/n, which is a rational quantity.