# What are surds and how does multiplying them work?

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A surd is an irrational number which cannot be expressed as a fraction or a recurring number. E.g. root 2 = 1.4142135…. and it continues (but it does NOT repeat like a recurring fraction though). E.g. pi would actually be a surd too, because it cannot be accurately presented as a simple number like a fraction or recurring decimal. 1.4.... or 32/9 are not surds for that matter.

If you were to round 1.4142135… to 1.414, and then square it when trying to get back to 2, you will in fact get 1.999396. This is because rounding is a type of approximation, so you will not get the exact figure for 2 but for a number close to it. This would be the same with all surds. Root 2 would be the only fully accurate representation of the number.

Hence (root 2)2 would be 2. To help understand this, imagine it was root 16.
Root 16= 4.
42 = 16.
Hence, (root 16)2 = 16.
In the same way, any rooted surd multiplied by itself = the number inside the surd.

And in terms of multiplying different surds…

Root 9 x Root 4 = Root 36

We can prove this is because

Root (9x4) = Root 36   = 6 and root 9 x root 4 = 3 x 2 = 6. Therefore, this is correct!

Hence, with any surd, the inside number of the surds can be multiplied to make a new surd. (e.g. root 3 x root 5 = root 15 ) :)

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