How to Simplify Surds

When learning how to simplify surds students need to understand the difference between a rational and irrational number.

Rational numbers include integers and terminating and repeating decimals. They can be written as a fraction with both the numerator and denominator as integers.

An irrational number is a number which, in its decimal form does not terminate or repeat.  This means it cannot expressed as a fraction with two integers.  π is one example of an irrational number which the students would have come across when learning about circles.

Another type of irrational number is a Surd.  A surd is a number written exactly using square or cube roots.  For example, √2 and 3√7 are surds.  √16 and 3√8 are not surds, because √16 = 4 and 3√8 = 2.

Rational and Irrational Numbers

To introduce surds students use their calculators to find the only irrational from a selection of rational numbers.  If the class need some help you could pose the question “Which of the following cannot be written as a fraction using two integers?”

Learning how to Simplify Surds

While it is often useful to approximate surds, for example, when finding the length of the hypotenuse in a right-angled triangle or the radius of a circle when given its area students need to be able to calculate with surds in their exact form.  This is needed when solving a quadratic equation by completing the square or using the formula.  Learning how to simplify surds is also needed for rationalising denominators.

A helpful exam tip I give to students is to keep a number in surd form until they get to the final answer.  This way they won’t carry through any rounding errors that would affect the final solution.

In the second slide we discuss how to simplify surds using the following examples.

Multiplying with Surds

√9 × √4 = 3 × 2 = 6

√9 × √4 = √(9 × 4) = √36 = 6

√12 = √4 × √3 = 2 × √3 = 2√3

Dividing with Surds

√100 ÷ √25 = 10 ÷ 5 = 2

√100 ÷ √25 =√4 = 2

As we work through some more examples I challenge the students to define these relationships.

How to Simplify Surds using the Multiplication Rule

How to Simplify Surds using the Division Rule

Later, I ask the class to attempt the following questions for themselves on their mini-whiteboards.


We work through each question one at a time using the learning from the previous problem to help progress with the next one.  When we have completed the third question students are ready to work through the problems on the third slide independently.  I encourage the class to check their answers using a calculator with a natural display.

Linking Surds to Algebra and Geometry

The plenary challenges students to link working with surds to setting up and solving equations involving the area of a rectangle and triangle. This should be attempted without the use of a calculator so the relationships we have learnt are applied.

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