Question 126164
Example 1



Consider this example



Add: {{{sqrt(4)+sqrt(16)}}}



{{{sqrt(4)+sqrt(16)}}} 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+4}}} 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(4)+sqrt(16)}}} simplifies to 6. In other words, {{{sqrt(4)+sqrt(16)=6}}}




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Example 2


Here's another example


Add: {{{sqrt(12)+sqrt(75)}}}




{{{sqrt(12)+sqrt(75)}}} 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*sqrt(3)+sqrt(75)}}} Simplify {{{sqrt(12)}}} to get {{{2*sqrt(3)}}}. Note: If you need help with simplifying square roots, check out this <a href=http://www.algebra.com/algebra/homework/Radicals/simplifying-square-roots.solver> solver</a>.



{{{2*sqrt(3)+5*sqrt(3)}}} Simplify {{{sqrt(75)}}} to get {{{5*sqrt(3)}}}.



Since we have the common term {{{sqrt(3)}}}, we can combine like terms


{{{(2+5)sqrt(3)}}} Combine like terms. Remember, {{{5x+3x-4x=(5+3-4)x=4x}}}



{{{7*sqrt(3)}}} Now add {{{2+5}}} to get {{{7}}}


So {{{sqrt(12)+sqrt(75)}}} simplifies to {{{7*sqrt(3)}}}. In other words,  {{{sqrt(12)+sqrt(75)=7*sqrt(3)}}}





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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.