SOLUTION: I need help simplifying this expression: {{{sqrt(24+8sqrt(5))}}} I removed the common factor out of the square root to obtain {{{2sqrt(6+2sqrt(5))}}}, but the answer key says it

Algebra ->  Radicals -> SOLUTION: I need help simplifying this expression: {{{sqrt(24+8sqrt(5))}}} I removed the common factor out of the square root to obtain {{{2sqrt(6+2sqrt(5))}}}, but the answer key says it       Log On


   

Question 1030250: I need help simplifying this expression:
sqrt%2824%2B8sqrt%285%29%29
I removed the common factor out of the square root to obtain 2sqrt%286%2B2sqrt%285%29%29, but the answer key says it is 2%2B2sqrt%285%29.
How is it possible? Am I missing out on a rule here?
Found 3 solutions by mananth, MathTherapy, greenestamps:
Answer by mananth(16949) About Me  (Show Source):
You can put this solution on YOUR website!
sqrt%2824%2B8sqrt%285%29%29
sqrt%284%2B8sqrt%285%29%2B20%29

sqrt%284%2B8sqrt%285%29%2B%282sqrt%285%29%29%5E2%29
%28a%2Bb%29%5E2+=+a%5E2%2B2ab%2Bb%5E2 using this identity
we get
sqrt%28%282%2B2sqrt%285%29%29%5E2%29

2%2B2sqrt%285%29

Answer by MathTherapy(10801) About Me  (Show Source):
You can put this solution on YOUR website!
I need help simplifying this expression:
sqrt%2824%2B8sqrt%285%29%29
I removed the common factor out of the square root to obtain 2sqrt%286%2B2sqrt%285%29%29, but the answer key says it is 2%2B2sqrt%285%29.
How is it possible? Am I missing out on a rule here?
*************************************
This happens to be one of those RARE cases when you can actually factor out a COMMON factor, which happens to be a
PERFECT SQUARE (in 4). This is what you did:
sqrt%2824+%2B+8sqrt%285%29%29 = sqrt%284%286+%2B+2sqrt%285%29%29%29 
                            sqrt%284%29+%2A+sqrt%286+%2B+2sqrt%285%29%29 ----- Applying sqrt%28m%2An%29 = sqrt%28m%29sqrt%28n%29         
                               2sqrt%286+%2B+2sqrt%285%29%29....At this point, you haven't fully simplified the
                                                       expression, and should've continued as follows:  
                            
                   2sqrt%285+%2B+1+%2B+2sqrt%285%2A1%29%29 ---- Changing 6 to 5 + 1, and 5 to 5*1  
                    2sqrt%285+%2B+1+%2B+2sqrt%285%29sqrt%281%29%29 ---- Applying sqrt%28m%2An%29 = sqrt%28m%29sqrt%28n%29         
   2sqrt%28%28sqrt%285%29%29%5E2+%2B+%28sqrt%281%29%29%5E2+%2B+2sqrt%285%29sqrt%281%29%29 --- Converting 
The above is in the form: %28a+%2B+b%29%5E2, with system%28matrix%282%2C3%2C+a%2C+being%2C+sqrt%285%29%2C+b%2C+being%2C+sqrt%281%29%29%29, and so:
2sqrt%28%28sqrt%285%29%29%5E2+%2B+%28sqrt%281%29%29%5E2+%2B+2sqrt%285%29sqrt%281%29%29 then becomes: 2sqrt%28%28sqrt%285%29+%2B+sqrt%281%29%29%5E2%29 
                                                                                2%28sqrt%285%29+%2B+sqrt%281%29%29 ----- Cancelling SQUARE and SQUARE ROOT
                                                                                2%28sqrt%285%29+%2B+1%29 = highlight%282sqrt%285%29+%2B+2%29
                                    This certainly matches the answer key: 2%2B2sqrt%285%29
=====
On the other hand, this author would've SIMPLIFIED the SURD from the onset, as follows:
                                 sqrt%2824+%2B+8sqrt%285%29%29
                       sqrt%2824+%2B+2%284%29sqrt%285%29%29 ----- Replacing 8 with factors, 2 & 4
                         sqrt%2824+%2B+2sqrt%2816%29sqrt%285%29%29 ----- Converting 4 to sqrt%2816%29
                              sqrt%2824+%2B+2sqrt%2880%29%29 ----- Applying sqrt%28m%29sqrt%28n%29 = sqrt%28mn%29%29
                        sqrt%2824+%2B+2sqrt%2820%2A4%29%29                
                    sqrt%2820+%2B+4+%2B+2sqrt%2820%29sqrt%284%29%29 ---- Changing 24 to 20 + 4, and applying sqrt%28m%2An%29 = sqrt%28m%29sqrt%28n%29           
  sqrt%28%28sqrt%2820%29%29%5E2+%2B+%28sqrt%284%29%29%5E2+%2B+2sqrt%2820%29sqrt%284%29%29 ---- Converting 
The above is in the form: %28a+%2B+b%29%5E2, with system%28matrix%282%2C3%2C+a%2C+being%2C+sqrt%2820%29%2C+b%2C+being%2C+sqrt%284%29%29%29, and so:
    sqrt%28%28sqrt%2820%29%29%5E2+%2B+%28sqrt%284%29%29%5E2+%2B+2sqrt%2820%29sqrt%284%29%29 then becomes:      sqrt%28%28sqrt%2820%29+%2B+sqrt%284%29%29%5E2%29
                                                                                                 sqrt%2820%29+%2B+sqrt%284%29 ----- Cancelling SQUARE and SQUARE ROOT
                                                                                                 highlight%282sqrt%285%29+%2B+2%29
                              This certainly matches the answer key: 2%2B2sqrt%285%29

Answer by greenestamps(13326) About Me  (Show Source):
You can put this solution on YOUR website!


Consider this patern:

%28sqrt%28a%29%2Bsqrt%28b%29%29%5E2=%28a%2Bb%29%2B2sqrt%28ab%29

Given an expression to be simplified in the form sqrt%28a%2Bb%2Asqrt%28c%29%29, if you can modify the expression so the coefficient "b" on the inside radical is 2, then the expression can be simplified using that pattern.

In your example, you can take "4" out of the coefficient "8", leaving the required coefficient "2"; the "4" goes back inside the radical as sqrt%2816%29:

sqrt%2824%2B8sqrt%285%29%29=sqrt%2824%2B2sqrt%2816%2A5%29%29=sqrt%2824%2B2sqrt%2880%29%29

The pattern now says you want to find integers a and b for which a+b=24 and ab=80. Those integers are 20 and 4, so