Rules of Surds

Rules of surds and examples can be found on this page.

When people (my students for instance) first see surds they say 'Ugh, we can't do these, THEY LOOK TOO HARD'. This is not uncommon, it is everybody's first impression of surds.

Surds look more difficult than they actually are. In fact, surds are one of the most easiest of sums to do. Once the concept is grasped and the general rules of surds are followed the math problem solving becomes easier to do.

A more mathematical definition of surds is:Surds are irrational numbers. They are numbers which cannot be written exactly.

Here are some general rules to follow regarding surds.

Rules of Surds

1. √a×√b=√ab

2. √a+√b=√a+b (Not True)

3. √b-√a=√b-a (Not True)

4. √a√b=√(ab)

5. √3b=3√b

6. 2√a=√4a

Tip to remember:






Multiplying Surds

Example 1

Work out: √2×√3

Solution: Using the rule above √a×√b=√ab. If we multiply the numbers 2 and 3 to give the answer √6.

Therefore √2×√3=√6

Example 2

Simplify: √12

Solution: using the rule above √a×√b=√ab we can break the number 12 up into smaller numbers such as √4×√3.

(NOTE: There are other ways to do this as 2 and 6 are also factors of 12, the final answer will still be the same)

Using one of the tips above it shows us that √4=2

So simplifying √4×√3 this further, to give us 2×√3

Next to make the answer look more complete and nicer we remove the multiplication sign and put the number 2 next to √3 to give


So, √12=2√3

Example 3

Work out 2√3×4√5

Solution: Multiply the numbers inside the square root with each other to give 3×5=15 and the outside numbers with each other to give 2×4=8.

Therefore the solution is 2√3×4√5=8√15

Dividing Surds

Example 4

Work out √15÷√5

Solution : Using this rule √a√b=√(ab)

Cancel the numbers down to its lowest form to give


Example 5

Simplify 8√6÷2√3

Solution : Numbers inside the square roots can be divided with each other, as well as the numbers on the outside. So 8÷2=4 and 6÷3=2

Therefore 8√6÷2√3=4√2

Adding Surds

Example 6

Work out √3+√6

Solution : Using the addition rule above √a+√b=√a+b, we can add the numbers 3 and 9 within the square root to give √9

Then, using the tip that √9=3

Therefore √3+√6=√9= 3

Subtracting Surds

Example 7

Work out √7-√2

Solution : Using the subtraction rule above √b-√a=√b-a, we can subtract the 7 and 2 to give 5

Therefore √7-√2=√5

Complex Questions on Surds

Example 8

Solve √-16

Solution can be viewed here on Twitpic, You need a twitter account to view it.

Example 9

Solve 2(x-1)²=-16

Solution can be viewed here on Twitpic, You need a twitter account to view it.

The main thing to remember about surds and working them out is that it is about manipulation. Changing and manipulating the equation so that you get the desired result. Rationalizing the denominator is all about manipulating the algebra expression.

Return from Rules of Surds to Math Glossary

Return from Rules of Surds to Math Problem Solving

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