Adding Square RootsFor example: 4√6 + 3√6 = ( 4 + 3 )√6 = 7√6 We see that both of our bases are 6, so the terms can be combined. We added our coefficients 4 and 3, and got a new coefficient of 7. If your bases do not match, first see if you can factor a perfect square out of one or both of your bases. If you cannot do this or if your bases still do not match, then you cannot work the problem. For example: √48 + 3√147 √( 16 x 3 ) + 3√( 49 x 3 ) 4√3 + (3 x 7)√3 4√3 + 21√3 ( 4 + 21 ) √3 25√3 Even though our problem did not begin with like bases, once we simplified we were able to combine the terms. Take care! Note our preexisting coefficient of 3 on the second term came down and had to be multiplied with 7 when we simplified √49. It is a common careless error to forget about this 3. Make sure to bring coefficients down with every step. Another example: 2√12 + 3√27y 2√( 4 x 3 ) + 3√( 9 x 3y) (2 x 4)√3 + (3 x 3)√3y 8√3 + 9√3y We cannot go any further, as our bases √3 and √3y do not match. With subtraction: 3√125y - √405y 3√( 25 x 5y ) - √( 81 x 5y ) (3 x 5)√5y - 9√5y 15√5y - 9√5y (15 - 9)√5y 6√5y After simplifying, we see that our bases match. When we combine, we make sure to find the difference, as this is a subtraction problem. Practice Problems Simplify 1. 3√10 + 2√10 2. √128 + √200 3. √99y - 2√44y 4. √243z + 2√245y Answers 1. (3 + 2)√10 = 5√10 2. √128 + √200 √( 64 x 2 ) + √( 100 x 2 ) 8√2 + 10√2 (8 + 10)√2 18√2 3. √99y - 2√44y √( 9 x 11y ) - 2√( 4 x 11y ) 3√11y - (2 x 2)√11y 3√11y - 4√11y ( 3 - 4 )√11y -1√11y 4. √243z + 2√245y √( 81 x 3z ) + 2√( 49 x 5y ) 9√3z + (2 x 7)√5y 9√3z + 14√5y Since our bases do not match, we cannot simplify further.
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