Manipulating Surds
Manipulating Surds
In this video – we’re going to be covering surds.
In short, surds are essentially expressions that produce an irrational number as a result of square rooting them.
A quick example would be comparing \(\sqrt{4}\) and $\sqrt{5}$ – we know that \(\sqrt{4}\) is equal to 2, so this wouldn’t be a surd. However, \(\sqrt{5}\) will not give us a whole or ‘rational’ number so this would be a surd.
Now we know what a surd is – we need to cover some rules you need to know when you are dealing with surds:
Rule 1: Adding Surds
When you see \(\sqrt{x} + \sqrt{y}\) this IS NOT the same as \(\sqrt{x+y}\). You cannot do anything like this when adding surds. This is the same for subtracting surds as well.
Rule 2: Multiplying Surds
Unlike addition and subtraction – when you see \(\sqrt{x} \times \sqrt{y}\), this CAN be made into \(\sqrt{x \times y}\).
For example, \(\sqrt{4} \times \sqrt{5} = \sqrt{4 \times 5} = \sqrt{20}\).
Rule 3: Dividing Surds
\(\frac{\sqrt{x}}{\sqrt{y}}\) is the exact same as \(\sqrt{\frac{x}{y}}\).
Rule 4: Rationalising Denominators
During your journey through surds, you’re going to come up against some pretty ugly looking questions where there is a surd showing as the denominator.
The way to get rid of this denominator is to do something called ‘Rationalising’.
To rationalise a denominator, you just need to multiply it by the surd you want to get rid of, but as a fraction equal to 1.
This sounds a little bit more complicated but I’ll illustrate it below. You need to remember that anything divided by itself will always equal 1.
Say we want to rationalise \(\frac{x}{\sqrt{y}}\) – we would do the following:
\(\frac{x}{\sqrt{y}} \times \frac{\sqrt{y}}{\sqrt{y}} = \frac{x \times \sqrt{y}}{\sqrt{y} \times \sqrt{y}} = \frac{x\sqrt{y}}{y}\)
As you can see, we multiplied our original fraction by \(\frac{\sqrt{y}}{\sqrt{y}}\) – remember, anything divided by itself is 1 so by doing this, we actually don’t change the value of the fraction. Obviously anything multiplied by 1 is just itself.
Now we’ve got a bit of a nicer denominator to deal with!
Hopefully, you found this helpful – below you will find some questions for you to practice what you’ve learned!
If you have any questions, please leave them in the comments below and I’ll get back to you as soon as possible.