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Revision:Surds

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(Redirected from Revision:C1 - Surds)

TSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics > Surds

Contents

What are surds?

A simple explanation is that a surd is the square root of a number which is not a perfect square. If you look in your textbook (why?) it will say that surds are roots that cannot be expressed as rational numbers. This means that most roots are surds: \sqrt{2}, \sqrt{3}, \sqrt{24}, the list goes on. You can try and work them out on a calculator, but that will only give you an estimate, accurate to ten decimal places or so - it's not exact. Surds are all "irrational" numbers - that means they can't be expressed as fractions, so their decimal expansions go on forever with no real pattern. (Fractions like \frac{1}{3} or \frac{1}{7} go on forever too, but they start repeating themselves after a while.)


Manipulations of surds

There are two basic identities you need to know.

  • \sqrt{a} \times \sqrt{b} = \sqrt{ab}. For example, \sqrt{2} \times \sqrt{3} = \sqrt{6}.
  • \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}. For example, \frac{\sqrt{20}}{\sqrt{5}} = \sqrt{\frac{20}{5}} = \sqrt{4}


IMPORTANT: There are no simple identities for adding and subtracting surds - in most cases, something like \sqrt{2}+\sqrt{5} can't be simplified!


Simplifying surds

For example:


\sqrt{72} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}


To simplify a surd \sqrt{x}, you have to find the largest perfect square that divides x. Above, that was 36. You then separate the two to get something of the form a\sqrt{b}. Sometimes this isn't easy - think of something like \sqrt{968}! If you can't immediately find the largest factor, then, it's a good idea to get rid of smaller factors to simplify the problem. For example:


\sqrt{968} = 2\sqrt{242} = 2 \times 11\sqrt{2} = 22\sqrt{2}


In extreme cases, just factorise the whole number (like in GCSE) and look for repeated factors:


[Unparseable or potentially dangerous latex formula. Error 6 ]


Hooray for calculators!


Rationalising the denominator

This is the tricky bit. When you're dealing with fractions, Edexcel hates it when you leave surds on the bottom - you have to "rationalise the denominator". For example:


\frac{1}{\sqrt{5}} = \frac{1}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{\sqrt{5}}{5}


See what we did there? We wanted to get rid of the surd from the denominator, so we multiplied top and bottom by the surd. That's basically it. Another example:


\frac{1}{2\sqrt{5}} = \frac{1}{2\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{\sqrt{5}}{2 \times \sqrt{5}^2} = \frac{\sqrt{5}}{10}


There are some fractions where this method won't work, though, because the denominator has more than one term in it. In that case, we have to use the difference of two squares. Take a look:


\frac{1}{\sqrt{2}+\sqrt{5}} = \frac{1}{\sqrt{2}+\sqrt{5}} \times \frac{\sqrt{2}-\sqrt{5}}{\sqrt{2}-\sqrt{5}} = \frac{\sqrt{2}-\sqrt{5}}{\sqrt{2}^2 - \sqrt{2}\sqrt{5} + \sqrt{2}\sqrt{5} - \sqrt{5}^2} = -\frac{\sqrt{2}-\sqrt{5}}{3}


The irrational bit cancels, making life easier for us all!


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