How to Simplify Surds: Explanation, Examples, and Questions
A surd is a special type of number that contains an irrational root. In our introduction to surds, we explored what surds are and when they’re used in mathematics.
Just like fractions, surds can sometimes be simplified to make calculations easier. When simplified, it’s easier to perform operations with surds, including addition, subtraction, multiplication, and division.
In this lesson, you’ll learn how to simplify surds, allowing you to handle calculations with them more easily.
What Does it Mean to Simplify Surds?
One of the first things you learn about fractions is how to simplify them.
Simplifying fractions makes it easier to perform calculations with them.
Surds can be simplified, meaning that they can be rewritten so that they have no square factors inside them.
How to Simplify Surds
To simplify surds we can use the product rule.
If a ≥ 0 and b ≥ 0 we have the following rules:
Product Rule:
\( \sqrt{ab} = \sqrt{a} \times \sqrt{b} \)
If \(a = b\), we also find that the square and square root cancel each other:
\(\sqrt{a} \times \sqrt{a} = \sqrt{a^2} = a\)
To simplify a surd, we first look for a square number that is a factor of the number inside the root.
Then, we can apply the product rule to separate the roots and simplify one of them.
- Find a square number that is a factor of the number inside the square root.
- Rewrite the surd using the product rule.
- Simplify the square root.
- Repeat this process if the root still has more square factors.
Eg. for √72, 72 = 36 × 2
√72 = √36 × √2
√36 × √2 = 6 × √2 = 6√2
In this example, 2 does not have any square factors so we are finished.
If this seems confusing don’t worry – we have plenty of examples!
Worked Examples
Example 1
Question: Simplify √50
Step 1: Find a square number that is a factor of 50.
The factors of 50 are:
- 1 and 50
- 2 and 25
- 5 and 10
25 is a square number.
Step 2: Rewrite the surd using the product rule.
√50 = √25 × √2
Step 3: Simplify the square root.
√25 × √2 = 5 × √2 = 5√2
Answer: √50 = 5√2
Example 2
Question: Simplify √28
Step 1: Find a square number that is a factor of 28.
The factors of 28 are:
- 1 and 28
- 2 and 14
- 4 and 7
4 is a factor of 28 which is also a square number.
Step 2: Rewrite the surd using the product rule.
√28 = √4 × √7
Step 3: Simplify the square root.
√4 × √7 = 2 × √7 = 2√7
Step 4: 7 does not have any more square number factors, so we are finished.
Answer: √28 = 2√7
Example 3
Question: Simplify √63
Step 1: Find a square number that is a factor of 63.
The factors of 63 are:
- 1 and 63
- 3 and 21
- 7 and 9
9 is a factor of 63 which is also a square number.
Step 2: Rewrite the surd using the product rule.
√63 = √9 × √7
Step 3: Simplify the square root.
√9 × √7 = 3 × √7 = 3√7
Step 4: 7 does not have any more square number factors, so we are finished.
Answer: √63 = 3√7
Example 4
Question: Simplify √48
Step 1: Find a square number that is a factor of 48.
The factors of 48 are:
- 1 and 48
- 2 and 24
- 3 and 16
- 4 and 12
- 6 and 8
4 is a factor of 48 which is also a square number.
Step 2: Rewrite the surd using the product rule.
√48 = √4 × √12
Step 3: Simplify the square root.
√4 × √12 = 2 × √12 = 2√12
Step 4: Notice that 12 still has a square factor of 4, so we repeat the process.
2√12 = 2 × √4 × √3
2√12 = 2 × 2 × √3 = 4√3
Step 5: 3 does not have any square factors, so we are finished.
Answer: √48 = 4√3
Now that you can simplify surds and expressions involving surds, the next step is to simplify surds in the denominators of fractions.
This is a skill called rationalising the denominator