How to Expand Brackets with Surds
Surd multiplication can also occur when multiplying brackets. When expanding brackets with surds, the process is exactly the same as without surds. You just need to remember the above rules for multiplying surds.
Single Brackets
In our lesson on multiplying single brackets we saw two methods for expanding single brackets: using a table or multiplying out mentally.
Using a Table
- If you can simplify any of the surds, simplify first.
- Enter the outside of the bracket in the first column and each term inside the bracket in the top row.
- Multiply the outside of the bracket by each term inside of the bracket.
- Simplify if you have any like terms.
Example 1
Question: Expand √2(4 + √3)
| × | 4 | + √3 |
| √2 | 4√2 | + √6 |
Answer: 4√2 + √6
Example 2
Question: Expand 5√2(4 – √72)
Simplify: 5√2(4 – √72) = 5√2(4 – √36√2) = 5√2(4 – 6√2)
| × | 4 | – 6√2 |
| 5√2 | 20√2 | – 60 |
Note: Calculation for the second term:
5√2 × -6√2 = 5 × -6 × √2 × √2 = -30 × (√2)2 = -30 × 2 = -60
Answer: 20√2 – 60
By Inspection Method
Once you become confident using a table method it can be faster to simply multiply by inspecting.
Be careful using this method as you are more likely to make mistakes.
Double Brackets
Double brackets can also be done in the same two methods we introduced in our lesson on how to multiply double brackets. We did this by using a table or by multiplying out mentally (sometimes called the FOIL method).
Now we’ll look at both methods with examples including surds.
Using a Table
Example 1
Question: Expand (6 + √5)(7 + 4√5)
| × | 6 | + √5 |
| 7 | 42 | + 7√5 |
| + 4√5 | + 24√5 | + 20 |
Note: Calculation for the last term:
4√5 × √5 = 4 × (√5)2 = 4 × 5 = 20
Simplify: 42 + 7√5 + 24√5 + 20 = 62 + 31√5
Answer: 62 + 31√5
Example 2
Question: Expand (3 + 2√6)(8 – 3√3)
| × | 3 | + 2√6 |
| 8 | 24 | + 16√6 |
| – 3√3 | – 9√3 | – 18√2 |
Note: Calculation for the last term:
2√6 × – 3√3 = 2 × -3 × √6 × √3 = -6√18 = -6√9√2 = -6 × 3 × √2 = -18√2
Answer: 24 + 16√6 – 9√3 – 18√2
After multiplying out the brackets we have to simplify by adding like surds together. In this example we have no like surds so it cannot be simplified.
FOIL Method
To multiply the brackets by inspection you have to perform 4 multiplications, add together the terms then simplify and collect like terms.
- Simplify the surds (if possible).
- Multiply the first terms of both brackets.
- Multiply the outer terms of both brackets.
- Multiply the inner terms of both brackets.
- Multiply the last terms of both brackets.
- Simplify and add together like surds (if possible).
Example 1
Example 2
Difference of Two Squares
The difference of two squares is the name for two brackets which are in the form: (a+b)(a-b). The value of a or b can also be a surd.
Difference of Two Squares Rule
In general, (a+b)(a-b) = a² – b²
Example 1
Question: Expand (3 + √7)(3 – √7)
This can be done with either method. Here we’ll do this using a table.
| × | 3 | + √7 |
| 3 | 9 | + 3√7 |
| – √7 | – 3√7 | – 49 |
= 9 + 3√7 – 3√7 – 49
Answer: -40
Example 2
Question: Expand (5 + √12)(5 – √12)
Simplify first: (5 + √12)(5 – √12) = (5 + 2√3)(5 – 2√3)
| × | 5 | + 2√3 |
| 5 | 25 | + 10√3 |
| – 2√3 | – 10√3 | – 12 |
Note: the last term is found by: 2√3 × -2√3 = 2 × -2 × (√3)2 = -4 × 3 = -12
(5 + 2√3)(5 – 2√3) = 25 + 10√3 – 10√3 – 12 = 13
Answer: 13
Now you know how to expand brackets with surds, use your knowledge to rationalise the denominator.
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