Year 10 Interactive Maths - Second Edition

Simplification of Surds

In general:

If a and b are positive numbers, then

Note:

Simplest Form of a Surd

A surd is said to be in its simplest form if the number under the root sign has no perfect square as a factor.

In general:

To simplify a surd, write the number under the root sign as the product of two factors, one of which is the largest perfect square.

Note that the factor 16 is the largest perfect square.

Recall that the numbers 1, 4, 9, 16, 25, 36, 49, … are perfect squares.

Example 4

Solution:

The square numbers are 1, 4, 9, 16, 25, 36, 49, ....  By trying each one in turn, we find that the largest perfect square that is a factor of 48 is 16.

Example 5

Solution:

Look for perfect squares that are factors of 63. Try 4, 9, 16 etc.

Alternative Method

Example 6

Solution:

Look for perfect squares that are factors of 12.  Try 4, 9, 16 etc.

Example 7

Key Terms

entire surd, mixed surd, simplest form, perfect squares