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Surds

A self-marking exercise on calculating, simplifying and manipulating surds (radicals)
  Menu   Level 1 Level 2 Level 3 Level 4 Level 5 Level 6 Level 7 Level 8 Exam Help
No calculator

Without using a calculator simplify the following. You can use the √ button to insert the root or radical symbol. You can earn a trophy if you get at least 9 questions correct and you do this activity online. Click the Help tab above for more information.

1

$$\sqrt{2}+\sqrt{32}$$

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2

$$\sqrt{12}+\sqrt{48}$$

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3

$$\sqrt{48}+\sqrt{108}$$

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4

$$\sqrt{80}+\sqrt{245}$$

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5

$$\sqrt{180}+\sqrt{245}$$

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6

$$\sqrt{343}+\sqrt{448}$$

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7

$$\sqrt{32}-\sqrt{8}$$

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8

$$8\sqrt{2}-2\sqrt{8}$$

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9

$$2\sqrt{27}+5\sqrt{12}$$

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10

$$20\sqrt{5}-5\sqrt{20}$$

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11

$$\sqrt{68}-\sqrt{17}$$

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12

$$\sqrt{96}+\sqrt{6}-\sqrt{24}$$

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Check

This is Surds level 4. You can also try:
Level 1 Level 2 Level 3 Level 5 Level 6 Level 7 Level 8

Instructions

Try your best to answer the questions above. Type your answers into the boxes provided leaving no spaces. As you work through the exercise regularly click the "check" button. If you have any wrong answers, do your best to do corrections but if there is anything you don't understand, please ask your teacher for help.

When you have got all of the questions correct you may want to print out this page and paste it into your exercise book. If you keep your work in an ePortfolio you could take a screen shot of your answers and paste that into your Maths file.
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Description of Levels

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Level 1 - Simplifying surds

Level 2 - Simplifying the product of two surds

Level 3 - Simplifying the product of integers and surds

Level 4 - Simplifying the sum of integers and surds

Level 5 - Simplifying fractions containing surds

Level 6 - Rationalising the denominator of a fraction

Level 7 - Miscellaneous questions involving surds

Level 8 - Finding the conjugate surd to rationalise the denominator

Paper Surprising Perimeter - A wonderful surds problem with a surprising result.

Exam Style Questions - A collection of problems in the style of GCSE or IB/A-level exam paper questions (worked solutions are available for Transum subscribers).

More on this topic including lesson Starters, visual aids, investigations and self-marking exercises.

Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don’t yet have a Transum subscription one can be very quickly set up if you are a teacher, tutor or parent.

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There are three statements referring to surds in the English National Curriculum:

Help

 

The key properties you need are:

Surds

Examples

$$\sqrt{80} = \sqrt{4\times4\times5} = \sqrt{4}\times\sqrt{4}\times\sqrt{5} = 2\times2\times\sqrt{5} = 4\sqrt{5}$$

$$\sqrt{8}\times\sqrt{12} = \sqrt{4\times2}\times\sqrt{4\times3} = 2\times\sqrt{2}\times2\times\sqrt{3} = 4\sqrt{6}$$

$$(5+\sqrt{7})(5-\sqrt{7}) = 25+5\sqrt{7}-5\sqrt{7}-7 = 25-7 = 18$$

 

Answers should be given in their simplest form. For example:

\( \sqrt{3}-1 \) is simpler than \( -1+\sqrt{3}\) as it uses less symbols.

\( \dfrac{5-\sqrt{3}}{2} \) is simpler than \( \dfrac{\sqrt{3}-5}{-2} \) as it uses less symbols.

 

Typing Mathematical Notation

These exercises use MathQuill, a web formula editor designed to make typing Maths easy and beautiful. Watch the animation below to see how common mathematical notation can be created using your keyboard.

MathQuill Animation

Don't wait until you have finished the exercise before you click on the 'Check' button. Click it often as you work through the questions to see if you are answering them correctly. You can double-click the 'Check' button to make it float at the bottom of your screen.

Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don’t yet have a Transum subscription one can be very quickly set up if you are a teacher, tutor or parent.

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