What is a surd?

A surd is a number (written in a special way) that cannot be simplified to remove a square root (or cube root etc).

What you normally see are typical numbers e.g. 4, 5, 4.5, -4, 5.

But, beyond that there are special types of numbers that you have to express with certain symbols. A surd is one of them.

A square root is a number where its value, when multiplied by itself, gives the resulting number.

Example: 4 × 4 = 16, so a square root of 16 is 4.

Going back to surds, we know that 2 x 2 = 4 and that 2 is a square root of 4.

So… if you then write √4, this can be simplified to 2. So √4 is not a surd. But √2 is a surd because you cannot further simplify it.

So the number √2 is a surd (that's how you express the number) but if you do (√2)² you get...2

So... they’re the basic rules you need to know for now.

You’ll learn more as we see it in action.

Let’s look at some questions now:

√(36) / √(72)= ?
a) √(1/2)
b) √(3/2)
c) √(12)
d) 2

To answer this question, we need to simplify these two square roots. A square root means the number which when multiplied by itself gives the required value - for instance, if 2 x 2 = 4, then √4 =2

By the same principle, since 6 x 6 = 36, √(36) = 6. Our equation now reads 6 ÷ √(72) - Remember, the division sign can be shown as / or ÷.

Next, we need to simplify √(72).

Since 72 = 2 x 36, we can rewrite this as √(36) x √2, or 6√2.

Our equation now reads 6 ÷ 6√2, and by cancelling the 6 above and below the line we get a result of √(1/2) (a).

Another way we can solve this question is by simplifying √(36) / √(72) to √(36/72). Since 36/72 = ½, this also simplifies to √(1/2) or 1/√2 (since √1= 1).

Here's another example:

Simplify √(72) - √(50)
a) √(22)
b) √(72/50)
c) 3√2
d) √2

This question similarly requires us to simplify these surds to their simplest form and then cancel any common factors.

We can start with √(72). Since this is equal to √(36)√(2), we know that it is then also equivalent to 6√2.

Then, we can simplify √(50).

In this case we know that √(25) = 5, so we can simplify √(50) to 5√2. Our equation then becomes 6√2-5√2.

Now that all our surds are the same base (√2), we can subtract them. Our expression reads (6-5)√2, which is the same as 1√2 or √2 (d).

Here's another practice question.

If a box is full of 125 smaller cubes with side length 2cm, what is the volume of the box?
a) 250 cm³
b) 1000 cm³
c) 125 cm³
d) 62.25cm³

This question involves a mixture of geometry and indices, and can be solved in two different ways - using volume, or using side length.

To solve this question using side length we need to consider the number of small boxes which fit along each side of the larger box. Since the volume of the large box is x³ where x = side length, the number of smaller boxes along each side is ∛(125)= 5 boxes.

Since each smaller box has a side length of 2cm, we therefore know that the side lengths of the larger cube are 5 x 2 = 10cm.

We can then cube this side length to get V=10³cm³ =1000cm³.

Another way to solve this question is to use volume. We know that the side length of each smaller cube is 2cm, meaning that its volume is 2³cm³ or 8cm³.

Since there are 125 smaller boxes in the larger box, we can then multiply this volume by the number of boxes to find the volume of the larger box:

125 x 8 = 1000

Both of these methods give us a volume of 1000 cm³ (b).

Key rules to remember

• When calculating make sure all surds have the same base
• Simplify all surds to their simplest form and cancel common

Practice time!

Now, it's your turn to practice.

Click on the button below and start your practice questions. We recommend doing untimed mode first, and then, when you're ready, do timed mode.

Every question has two solutions videos after you complete the question. The first is a quick 60 second video that shows you how our expert answers the question quickly. The second video is a more in-depth 5-steps or less explainer video that shows you the steps to take to answer the question. It's really important that you review the second video because that's where you'll learn additional tips and tricks.

Once you're done with the practice questions, move on to the next checkpoint.

Now, let’s get started on your practice questions.

---