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Mr Barton’s Maths Notes Number 10. Surds
10. Surds What on earth is a surd and why do we need them?… Let’s get it out of the way before we start… yes, I know, “surd” sounds a little bit like a rude word, and my pupils never tire of reminding me of that every lesson… What are they?... Surds are just numbers left in square-root form, like or But why do we need them?... Because such numbers are irrational, and if we tried to write them out as decimals, they would go on forever! The Two Important Rules of Surds Everything we are going to look at in this section is based around these two crucial rules: Rule 1 If you have a surd and you multiply it by another surd, then the answer is just the same as the surd of the original two numbers (a and b) multiplied together e.g. Rule 2 If you multiply a surd by itself, then the answer is just the original number before it was square-rooted e.g.
1. Simplifying Single Surds Okay, this is probably the nicest type of surd question you could get asked. You need to make the number under the square root sign as small as possible And it’s nice and easy so long as you know your square numbers! Method 1. Split up the number being square-rooted into a product of at least one square number 2. Use Rule 1 to simplify your answer Remember: Square Numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100… Example 1 Simplify: Okay, so we need to split up 50. We ask ourselves: “which square number is a factor of 50?” Well, if you look along the list above, you should notice that… 25 is! So, using Rule 1… Now, because we’ve chosen a square number, that’s going to simplify nicely… So… Example 2 Simplify: Okay, so this time we need to split up 45. We ask ourselves: “which square number is a factor of 45?” Well, if you look along the list above, you should notice that… 9 is! So, using Rule 1… Now, because we’ve chosen a square number, that’s going to simplify nicely… So…
2. Simplifying more than one Surd (Multiplying) Again this is fairly easy so long as you could understand the previous section Method 1. Deal with each surd individually 2. Split up the numbers being square-rooted into a product of at least one square number 3. Use Rule 1 to simplify your answers 4. When simplifying the whole answer, treat your whole numbers and surds separately Remember: Square Numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100… Example Simplify: Okay, let’s deal with each surd individually and split them up exactly like we did in the previous section: So… To simplify further we multiply our whole number and our surds separately and…So… And if you wanted to be really clever, we can simplify even further… So…
3. Simplifying more than one Surd (Dividing) Good News: Do these in exactly the same way as the Multiplying ones! Example Simplify: Okay, let’s deal with each surd individually and split them up because we’re good at that!... So… Let’s sort out the multiplication on the top line like we did before… and…So… So (after what seems like ages) we are left with: So… But wait a minute! We can use division to simplify, just like we used multiplication… and… So… But we can be clever again and go a wee bit further…
4. Simplifying more than one Surd (Adding and Subtracting) Just like when we are adding and subtracting fractions, there is a little twist!... Twist We can only add and subtract surds of the same type So… we must use our simplifying skills to change them into the same type! Example 1 Simplify: Now, one thing is for certain: the answer is definitely NOT: No way! No such rule! Don’t forget! We need to simplify the surds to see if that helps So… Look, our surds are now of the same type! They are both: So we can now just add our whole numbers, because 2 lots of something, plus 3 lots of something must equal 5 lots of something! So we have our answer… Example 2 Simplify: Now, one thing is for certain: the answer is definitely NOT: No way! No such rule! Don’t forget! We need to simplify the surds to see if that helps So… Look, our surds are now of the same type! They are both: So we can now just subtract our whole numbers, because 3 lots of something, minus 2 lots of something must equal 1 lot of something! So we have our answer…
5. Rationalising the Denominator! Warning: This is hard, and should only be attempted by the very brave… What does Rationalising the Denominator mean?... Basically, it is considered a bit untidy in the fussy world of mathematics to have a surd on the bottom of a fraction (the denominator). So, if we can get rid of all the surds off the bottom of a fraction, we get rid of all the irrational numbers, and so we rationalise the denominator! Method Multiply the top and the bottom of the fraction by the same carefully chosen expression! Example 1 – Single Surd Rationalise the denominator of: Okay, so we don’t like the look of that on the bottom What could we multiply it by to make it disappear?... Well, using Rule 2… how about by itself! Be careful: Remember, whatever we multiply the bottom of the fraction by, we must also do to the top, otherwise the value of the fraction changes, so we will have changed the question! Using our Rules of Fractions, we just multiply the tops together, and then the bottoms together And using Rule 2… So, we are left with our answer! And if you check them on the calculator, you will see they give the same answer!
Example 2 – Surd with Other Numbers Rationalise the denominator of: Trick For questions like this, the thing you multiply both the top and the bottom of the fraction by is just the expression on the bottom, but with the sign changed! Why, I hear you ask?... Well, it’s all to do with the difference of two squares… Again, we multiply tops and bottoms together, but we also use our methods of expanding brackets (see the Algebra section) So… our answer must be… Okay, so let’s multiply the top and the bottom of the fraction by… change the sign… And if you check them on the calculator, you will see they give the same answer! Tops Bottoms Use FOIL Now look what happens when we collect up our terms and simplify The middle two terms cancel out, and we are left with a very nice (and rationalised) denominator!