SOLUTION: Why is it important to simplify radical expressions before adding or subtracting?

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Question 126164: Why is it important to simplify radical expressions before adding or subtracting?
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Example 1


Consider this example


Add: sqrt%284%29%2Bsqrt%2816%29


sqrt%284%29%2Bsqrt%2816%29 Start with the given expression. Can you add these two radicals in this state? The answer is simply no. We need to simplify the radicals.


2%2B4 Take the square root of 4 to get 2. Take the square root of 16 to get 4. In this step we have simplified the radicals.

6 Add


So sqrt%284%29%2Bsqrt%2816%29 simplifies to 6. In other words, sqrt%284%29%2Bsqrt%2816%29=6



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Example 2

Here's another example

Add: sqrt%2812%29%2Bsqrt%2875%29



sqrt%2812%29%2Bsqrt%2875%29 Start with the given expression. Once again, is it possible to add these without any simplification? Unfortunately, you cannot. So we have to simplify.


2%2Asqrt%283%29%2Bsqrt%2875%29 Simplify sqrt%2812%29 to get 2%2Asqrt%283%29. Note: If you need help with simplifying square roots, check out this solver.


2%2Asqrt%283%29%2B5%2Asqrt%283%29 Simplify sqrt%2875%29 to get 5%2Asqrt%283%29.


Since we have the common term sqrt%283%29, we can combine like terms

%282%2B5%29sqrt%283%29 Combine like terms. Remember, 5x%2B3x-4x=%285%2B3-4%29x=4x


7%2Asqrt%283%29 Now add 2%2B5 to get 7

So sqrt%2812%29%2Bsqrt%2875%29 simplifies to 7%2Asqrt%283%29. In other words, sqrt%2812%29%2Bsqrt%2875%29=7%2Asqrt%283%29




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Summary:
So the reason we have to simplify the radicals is we can't combine two radicals that have different radicands (the values in the square roots). So when we simplify, we find that the two roots might have a common root. If that's the case, we can combine them.