Consider an example of adding two radical expressions: `sqrt(3)` and `sqrt(12)` .
If we attempt to simplify `sqrt(3) + sqrt(12)` , as in any calculation of a set of arithmetic operations, order of operation convention must be applied. Recall that radical is considered a grouping symbol: it must be evaluated first, before any other operation. Which means we cannot add `sqrt(3) ` and `sqrt(12)` easily and get an exact answer, because both of these radicals are non-terminating, non-repeating decimals.
However, since there is a property of radicals called "product rule", which states that the product of radicals is a radical of a product, `sqrt(12)` can be simplified:
`sqrt(12) = sqrt(4*3) = sqrt(4) * sqrt(2) = 2sqrt(3)`
Now we have to add `sqrt(3) ` and `2sqrt(3)` . This is possible because of the distributive property:
`1*sqrt(3) + 2*sqrt(3) = (1+2)sqrt(3) = 3sqrt(3)`
We apply the same property when we add polynomials and combine like terms. For example, `2x + 3x = (2 + 3)x = 5x` .
In this way addition of radical and polynomial expressions is similar. In fact, radicals such as `sqrt(3) and 2sqrt(3)` , and any other radical that can be simplified to become a multiple of `sqrt(3)` , are called like radicals.
Radicals that are not like cannot be combined, for example, one cannot simplify `sqrt(2) + 3sqrt(5)` , much like one cannot simplify `x + 3x^2` .
Radical expression to simplify:
`3sqrt(45) - 2sqrt(20)`