A surd is just a special type of number with irrational roots. In our introduction to surds lesson, we looked at what a surd is and when we might use surds.
Just like fractions, surds can sometimes be simplified. They can be written in a way which makes it easier to perform surd operations such as addition, subtraction, multiplication, and division.
In this lesson you’ll learn how to simplify surds so that you can easily perform calculations with surds.
Simplifying Surds
What Does it Mean to Simplify Surds?
One of the first things you learn about fractions is how to simplify them.
For example, 4/8 is the same as 1/2 and 6/9 = 2/3. It’s easier perform addition and multiplication etc. on simplified fractions so it’s better to simplify them first.
Surds can be simplified, meaning that they can be written in a way which makes it easier to perform calculations on them.
How to Simplify Surds
To simplify surds we need to use these general surd laws.
Product Rule:
If a = b we also find that the square and square root cancel each other:
Quotient Rule:
If we want to simplify a surd we need to find a square number that is a factor of the number inside the root sign and apply the above surd laws.
- Find a square number that is a factor of the number inside the square root. Eg. in √72, 72 = 36 × 2 where 36 is a square number.
- Rewrite the surd using the product rule. Eg. √72 = √36 × √2.
- Calculate the value of this square root. Eg. √36 = 6. So √36 × √2 = 6 × √2 = 6√2
- Repeat this process if the root still has more square factors. In the case of √2, 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 factor of 50 which is also a square number.
Step 2: Rewrite the surd using the product rule.
√50 = √25 × √2
Step 3: Calculate the value of this square root.
√50 = √25 × √2 = 5 × √2 = 5√2
Step 4: Repeat this process if the root still has more square factors.
2 does not have any more square number factors so we are finished
Answer: √50 = 5√2
Example 2
Question: Simplify √28
Step 1: 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: √28 = √4 × √7
Step 3: √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
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.
√63 = √9 × √7 = 3 × √7 = 3√7
7 does not have any more square number factors so we are finished.
Answer: √63 = 3√7
Example 4
Question: Simplify √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.
√48 = √4 × √12
√48 = √4 × √12 = 2 × √12 = 2√12
However, this example is incomplete, as 12 also has a square factor of 4.
Repeating the process:
2√12 = 2 × √4 × √3
2√12 = 2 × 2 × √3
2√12 = 4 × √3 = 4√3
3 does not have any square factors so now we have fully simplified the surd.
Answer: √48 = 4√3
In this last example we had to simplify twice as √48 = 2√12 was not completely simplified. If we look at the factors of 48 again, we see that 16 is also a square factor of 48.
√48 = √16 × √3 = 4√3
If we had chosen 16 as the square factor we would have only simplified one time.
Note: If there is more than one square factor, always choose the largest factor. This will eliminate the need to repeat the process.
Simplifying Surds Using Prime Factorisation
In the last example, we saw how there can be extra steps if a smaller square number is chosen.
Using this method can also be difficult when we have to simplify a surd includes a large number or a number with many factors.
Using prime factorisation can also help in simplifying surds.
- Step 1: Write the surd as a product of primes. Eg. √72 = √(2 × 2 × 2 × 3 × 3).
- Step 2: Write any repeated factors as square numbers. Eg. √(2 × 2 × 2 × 3 × 3) = √(2 × 4 x 9).
- Step 3: Apply the product rule and simplify. Eg. √(2 × 4 x 9) = √2 × √4 x √9 = √2 × 2 × 3
- Step 4: Multiply the factors and simplify. Eg. √2 × 2 × 3 = √2 × 6 = 6√2
Worked Examples
Example 1
Simplify √450
Using prime factorisation we find that 450 = 2 × 3 × 3 × 5 × 5
Rewrite √450 as √(2 × 3 × 3 × 5 × 5)
Write repeated factors as square numbers:
√(2 × 3 × 3 × 5 × 5) = √(2 × 9 × 25)
Apply the product rule and simplify:
√(2 × 9 × 25) = √2 × √9 × √25
= √2 × 3 × 5
= √2 × 15
= 15√2
Example 2
Simplify √600
Using prime factorisation we find that 600 = 2 × 2 × 2 × 3 × 5 × 5
Rewrite √600 as √(2 × 2 × 2 × 3 × 5 × 5)
Write repeated factors as square numbers:
√(2 × 2 × 2 × 3 × 5 × 5) = √(2 × 4 × 3 × 25)
Apply the product rule and simplify:
√(2 × 4 × 3 × 25) = √2 × √4 × √3 × √25
= √2 × 2 × √3 × 5
= 2 × 5 × √2 × √3
= 10√6
Note: This last step requires multiplying surds. Ie. √2 × √3 = √6.
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