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By the end of this topic, you should be able to:
A surd is the square root of a number that doesn't work out to be a whole number. In other words, it's a square root that cannot be simplified to give you an exact answer without a root sign.
Examples of surds:
Not surds:
Surds are useful because they let you give exact answers instead of decimal approximations. For example, √2 is exact, but 1.414213... goes on forever and is only approximate.
To simplify a surd, you need to find the largest square number that divides into the number under the square root. Then you can split the surd into two parts.
The rule: √(a × b) = √a × √b
This means you can break apart what's under the square root sign.
Step 1: Find the largest square number that is a factor of the number under the root.
Common square numbers to remember: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144...
Step 2: Split the surd into the square number part and the remaining part.
Step 3: Take the square root of the square number and write it in front.
Example 1: Simplify √18
Example 2: Simplify √20
Example 3: Simplify √180
Example 4: Simplify √27
You can only add or subtract surds that are the same type. This is just like collecting like terms in algebra.
The rule: Only surds with the same number under the root can be combined.
Example 1: 3√2 + 7√2
Example 2: 12√3 − 4√3
Example 3: 12√5 + 3√5
Example 4: 2√11 − 7√11
What you CANNOT do:
You cannot add or subtract different surds directly.
For example: 2√3 + 4√6 cannot be simplified because √3 and √6 are different surds.
Example: 2√8 + 5√2
Example from syllabus: √200 − √32
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