Adding+and+Subtracting+Radicals

Adding and Subtracting Radicals is similar to adding and subtracting polynomials because when adding and subtracting polynomials you can only add and subtract like terms that have the same variable and exponent and when adding and subtracting radicals **you can only subtract terms that have the same number in the radical.**

For example, you CAN combine the following radicals:



You CANNOT combine these radicals:



**TASK: Give an example of three radicals that you could add or subtract. Explain how you know that you could combine them.**


 * How do we combine radicals that DO have the same number in the radical?**

We combine radicals that have same number in the radical in the same way we combine like terms with the same variable and the same exponent. We add and subtract like terms by adding or subtracting the coefficients and keep the same variable (for example, 2x + 3x = 5x). We do the same thing with radicals. For example, if we have:



As shown above, we add the coefficients and we keep the radical the same (as long as both terms have the same number in the radical).

We do the same thing to subtract radicals with the same number under the radical except instead of adding the coefficients we subtract them. For example:




 * How do we add or subtract radicals that DO NOT have the same number in the radical?**

We know that we can only add or subtract radicals that have the same number under the radical. So, how would we simplify something like:

?

The first thing we would have to do before we could combine the terms is to reduce the radicals (like we did in the previous lesson) so that they have the same number under the radical.

We reduce the first radical as follows:

We can't reduce the second radical because there are no perfect square factors of 5.

Our new problem becomes: and now they have the same number under the radical, so we can subtract them and we get:



**TASK: Compare and contrast how adding and subtracting radicals with the same number in the radical and adding and subtracting radicals with different numbers in the radical. Give at least one similarity and one difference.**

**TASK: Open the document and complete the practice problems. **