If you know how to multiply out a double bracket containing algebra then expanding brackets containing radicals can be done in similar way. However, you will need to know some of the basic rules involving surds which are:

√x × √y = √(xy)

√x × √x = x

a√x × c√y = ac√(xy)

Example 1

Work out (3 + √5)(2 + √5)

First multiply the first term in the first bracket by the first term in the second bracket:

3 × 2 = 6.

Secondly, multiply the first term in the first bracket by the second term in the second bracket:

3 × √5 = 3√5

Thirdly, multiply the second term in the first bracket by the first term in the second bracket:

√5 × 2 = 2√5

Lastly, multiply the second term in the first bracket by the second term in the second bracket:

√5 × √5 = 5

So if you put all of these answer together on one line you get:

6 + 3√5 + 2√5 + 5

This can now be simplified further to:

11 + 5√5

Example 2

Work out (2 - 3√7)(4 - 2√7)

First multiply the first term in the first bracket by the first term in the second bracket:

2 × 4 = 8

Secondly, multiply the first term in the first bracket by the second term in the second bracket:

2 × -2√7 = -4√7

Thirdly, multiply the second term in the first bracket by the first term in the second bracket:

-3√7 × 4 = -12√7

Lastly, multiply the second term in the first bracket by the second term in the second bracket:

-3√7 × -2√7 = 6 × 7 = 42

So if you put all of these answer together on one line you get:

8 - 4√7 - 12√7 + 42

This can now be simplified further to:

50 - 16√7

Example 3

Work out (7 + 3√6)(4 + 2√5)

First multiply the first term in the first bracket by the first term in the second bracket:

7 × 4 = 28

Secondly, multiply the first term in the first bracket by the second term in the second bracket:

7 × 2√5 = 14√5

Thirdly, multiply the second term in the first bracket by the first term in the second bracket:

3√6 × 4 = 12√6

Lastly, multiply the second term in the first bracket by the second term in the second bracket:

3√6 × 2√5 = 6√30

So if you put all of these answer together on one line you get:

28 + 14√5 + 12√6 + 6√30

This cannot be simplified any further as all of the surds are different.