SOLUTION: I have been struggling to figure out this math problem and I was wondering if someone could help me? Please and Thank You!! I would deeply appreciate it!! Binomials Containing R

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: I have been struggling to figure out this math problem and I was wondering if someone could help me? Please and Thank You!! I would deeply appreciate it!! Binomials Containing R      Log On


   

Question 178266This question is from textbook Algebra and Trigonometry Structure and Method book 2
: I have been struggling to figure out this math problem and I was wondering if someone could help me? Please and Thank You!! I would deeply appreciate it!!
Binomials Containing Radicals
Simplify. Assume that each radical represents a real number.
(√(a-√a) *√(a+√a) )/( √(a-1))

This question is from textbook Algebra and Trigonometry Structure and Method book 2

Found 2 solutions by Alan3354, jim_thompson5910:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Simplify. Assume that each radical represents a real number.
(√(a-√a) *√(a+√a) )/( √(a-1))
-----------------
sqrt%28a+-+sqrt%28a%29%29%2Asqrt%28a+%2B+sqrt%28a%29%29%2Fsqrt%28a-1%29
=sqrt%28%28a+-+sqrt%28a%29%29%2A%28a+%2B+sqrt%28a%29%29%29%2Fsqrt%28a-1%29
=%28sqrt%28a%5E2+-+a%29%29%2F%28sqrt%28a-1%29%29
=%28sqrt%28a%29%2Asqrt%28a+-+1%29%29%2F%28sqrt%28a-1%29%29
=sqrt%28a%29

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
First, let's simplify the numerator sqrt%28a-sqrt%28a%29%29%2Asqrt%28a%2Bsqrt%28a%29%29

Let x=sqrt%28a%29

sqrt%28a-sqrt%28a%29%29%2Asqrt%28a%2Bsqrt%28a%29%29 Start with the given expression.


sqrt%28a-x%29%2Asqrt%28a%2Bx%29 Plug in x=sqrt%28a%29


sqrt%28%28a-x%29%28a%2Bx%29%29 Combine the roots using the identity sqrt%28A%29%2Asqrt%28B%29=sqrt%28A%2AB%29


sqrt%28a%5E2-x%5E2%29 FOIL


sqrt%28a%5E2-%28sqrt%28a%29%29%5E2%29 Plug in x=sqrt%28a%29


sqrt%28a%5E2-a%29 Square sqrt%28a%29 to get "a"


sqrt%28a%28a-1%29%29 Factor out the GCF "a"


sqrt%28a%29%2Asqrt%28a-1%29 Break up the square root using the identity sqrt%28A%2AB%29=sqrt%28A%29%2Asqrt%28B%29 (this just the reverse of the previous identity)


So the numerator sqrt%28a-sqrt%28a%29%29%2Asqrt%28a%2Bsqrt%28a%29%29 simplifies to sqrt%28a%29%2Asqrt%28a-1%29
------------------------------------------------


Now let's go back to the main problem:


So the expression


%28sqrt%28a-sqrt%28a%29%29%2Asqrt%28a%2Bsqrt%28a%29%29%29%2F%28sqrt%28a-1%29%29


simplifies to


%28sqrt%28a%29%2Asqrt%28a-1%29%29%2F%28sqrt%28a-1%29%29 (see steps above)



%28sqrt%28a%29%2Ahighlight%28sqrt%28a-1%29%29%29%2F%28highlight%28sqrt%28a-1%29%29%29 Highlight the common terms.


%28sqrt%28a%29%2Across%28sqrt%28a-1%29%29%29%2F%28cross%28sqrt%28a-1%29%29%29 Cancel out the common terms.


sqrt%28a%29 Simplify


==========================================================

Answer:


So %28sqrt%28a-sqrt%28a%29%29%2Asqrt%28a%2Bsqrt%28a%29%29%29%2F%28sqrt%28a-1%29%29=sqrt%28a%29 where a%3E1