SOLUTION: Simplify (89 + 28√10)^(¼) = √a + √b

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Question 1209554: Simplify
(89 + 28√10)^(¼) = √a + √b
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!
.
Simplify (89 + 28√10)^(¼) = √a + √b.
~~~~~~~~~~~~~~~~~~~ 

First, it is easy to see that

     = .    (1)


Indeed, if you square right side of (1), you will get  .


Thus, equality (1) is proved.



Second,

     =  + .    (2)


Indeed, if you square right side of (2), you will get  .


Thus, equality (2) is proved.



Equalities (1) and (2) combined mean that 

     =  + .


QED.     So,  a= 5,  b= 2.

Solved.

This problems teaches to extract  MENTALLY  square roots of the form
 with integer  x,  y,  z,  as they are in this assignment.



Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!


Consider this general case:





Turning that operation around, if we need to evaluate



then the requirement is to find a and b such that p = a+b and q = ab.

Example 1:

a+b=8; ab=15 --> a and b are 3 and 5 -->

Example 2: -->

a+b=16; ab=60 --> a and b are 6 and 10 -->

Now for your problem....

We are to find the 4th root of the given expression, so we will use the process described above to find the square root of the given expression and use the process again to find the 4th root.







a+b=89; ab=1960 --> a and b are 40 and 49 -->

Now find

a+b=7; ab=10 --> a and b are 2 and 5 -->

ANSWER:


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