Reducing Surds

A surd is a number that has a square root that cannot be simplified into a whole number or a simple fraction.

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Examples of Surds: $\sqrt{2}$ , $\sqrt{3}$ , $\sqrt{5}$

Non-Surds: $\sqrt{4} = 2$ (because it simplifies to a whole number) Surds are important in maths because they often show up in geometry, algebra, and other areas, so understanding how to work with them is essential.

Three Important Rules for Surds

Adding and Subtracting Surds

You can only add or subtract surds that have the same number inside the square root.

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Example 1: $3\sqrt{5} + 2\sqrt{5} = 5\sqrt{5}$

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Example 2: $4\sqrt{3} + 2\sqrt{2}$ cannot be simplified because $\sqrt{3}$ and $\sqrt{2}$ are different.

Multiplying Surds

When you multiply surds, multiply the numbers inside the square roots together.

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Example: $\sqrt{2} \times \sqrt{3} = \sqrt{2 \times 3} = \sqrt{6}$

Dividing Surds

When you divide surds, divide the numbers inside the square roots.

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Example: $\frac{\sqrt{18}}{\sqrt{2}} = \sqrt{\frac{18}{2}} = \sqrt{9} = 3$

Simplifying Surds

Simplifying a surd means breaking it down into a simpler form using the rules above.

Steps to Simplify a Surd:

Break Down the Number: Start by breaking the number inside the square root into factors (smaller numbers that multiply together to give the original number).
Look for Perfect Squares: Check if any of these factors are perfect squares (like $4, 9, 16, 25$ ), because you can simplify their square roots.

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Example: Simplifying $\sqrt{72}$

Break down $72$ :

$72 = 36 \times 2$
We pick 36 because it's a perfect square.

Apply the square root:

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

Simplify:

$\sqrt{36} = 6$ , so $\sqrt{72} = 6\sqrt{2}$ Final Answer: $\sqrt{72} = 6\sqrt{2}$

Combining Simplified Surds

After simplifying surds, you can add or subtract them if they are the same type.

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Example: Simplifying and Adding Surds Problem: Simplify $\sqrt{50} + \sqrt{18}$

Simplify each surd separately:

$\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$
$\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$

Add the surds:

Since both surds are now $\sqrt{2}$ , you can add them:
$5\sqrt{2} + 3\sqrt{2} = 8\sqrt{2}$ Final Answer: $\sqrt{50} + \sqrt{18} = 8\sqrt{2}$

Recap

Simplifying surds helps make them easier to work with.
Adding/subtracting is only possible when the numbers inside the square roots are the same.
Multiplying/dividing surds is straightforward by applying the rules. With these notes, you now have a clear path to mastering surds. Keep practicing to build your confidence and improve your skills!

Reducing Surds Simplified Revision Notes for Junior Cycle Mathematics

Reducing Surds

Three Important Rules for Surds

Simplifying Surds

Combining Simplified Surds

Recap

500K+ Students Use These Powerful Tools to Master Reducing Surds For their Junior Cycle Exams.

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