Explain how the process of combining radicals through addition and subtraction is similar to combining polynomials. What makes two radicals like radicals? Please give example.
Adding and subtracting radicals is similar to adding and subtracting polynomial terms. If radicals are same, we can add or subtract "like terms" in a similar way to polynomials.
For eg. `5x^3+4x^3=9x^3` or `9x^2-2x^2=7x^2` .In both cases the operations are possible due to like terms i.e. the terms whose variables and exponents are same but coefficients may be different.
We follow the same rule in case of combining radicals.
Two radicals are like radicals if they have the same index and same radicand but coefficients may be different. Eg. `3sqrt(x)` and `12sqrt(x)` are like radicals.
`5sqrt(2)+4sqrt(2)` is a sum of two radicals, and they can be added to produce `9sqrt(2)` . Again,`6root(3)(5)-2root(3)(5)=4root(3)(5)` .
Again for example: `sqrt(18)-2sqrt(27)+3sqrt(3)-6sqrt(8)`
Here we have to simplify and the combine like radicals.