The product of two unlike surds cannot be rational. Let and be unlike surds, then × cannot be rational number. In like manner, the quotient of two unlike surds cannot be rational. Let and be unlike surds, then cannot be rational number. In this sub-topic, we will learn the product and quotient of surds.

Product of surds

In multiplying surds, surds are combined using multiplication rule to write as a single surd, simplified where possible.

When simplifying surds such as  where we use the fact that  This can be used in reverse to simplify the product of two surds.

We know that: 

So, multiplying surds that have the same number inside the square root gives a rational number.

Suppose we were asked to simplify the expression

which cannot be simplified any further.
When multiplying surds, we simply multiply the numbers inside the square roots.

Key Point

• The following formulas may be used to simplify expressions involving surds:

Quotient of surds

In dividing surds, surds are combined using division rule to write them as a single surd, simplified where possible.

Suppose we were asked to simplify the

which cannot be simplified any further.

When dividing surds, we simply divide the numbers inside the square roots.

Key Point

The following formulas may be used to simplify expressions involving surds: