Calculus - Reiny, Friday, December 23, 2011 at 6:59pm
I read that as
(4a^2-1)/(4a^2-16) * (2-a)/(2a-1) , my brackets are necessary
= (2a+1)(2a-1)/( 4(a+2)(a-2) * (2-a)/(2a-1)
the (2-a)/(a-2) are opposite, they will give you the -1
so..
= -(2a+1)/(4(a+2)) , again you have to use brackets to write it on here.
Calculus - -Untamed-, Saturday, December 24, 2011 at 5:17pm
Thanks Reiny =) But I to get everything clear, the 2-a and the a-2 cancel right? After I cancel those I am left with (2a+1) as the numerator:S Sorry I am just confused. Also did you factor out the 2a+4 and 2a-4 to get 4(a+2)(a-2)? Could you please explain step by step?
(2a+1)(2a-1)/( 4(a+2)(a-2) * (2-a)/(2a-1)
(2a+1)(2a-1) (2-a)
====================
4(a+2)(a-2)(2a-1)
(2a+1)(2a-1) (2-a)
====================
4(a+2)(-1)(2-a)(2a-1)
-(2a+1)(1) (2-a)
====================
4(a+2)(2-a)(1)
-(2a+1)(1) (1)
====================
4(a+2)(1)(1)
-(2a+1)
====================
4(a+2)
Thanks Damon =)
To simplify the expression (4a^2-1)/(4a^2-16) * (2-a)/(2a-1), let's go through the steps:
Step 1: Factor both the numerator and denominator of each fraction:
The first fraction (4a^2-1)/(4a^2-16) can be factored as (2a+1)(2a-1) / (2a+4)(2a-4).
The second fraction (2-a)/(2a-1) does not need further factoring.
Step 2: Rewrite the expression using the factored form:
(2a+1)(2a-1)/(2a+4)(2a-4) * (2-a)/(2a-1).
Step 3: Simplify:
If we look at the terms (2a-1) and (2-a) in the numerator and denominator, we see that they are opposites of each other and will cancel out.
So, we can simplify the expression to -(2a+1)/(2a+4).
To clarify the steps taken:
- To factor the first fraction, we used the difference of squares formula, which is a^2 - b^2 = (a+b)(a-b). In this case, a is 2a and b is 1, resulting in (2a+1)(2a-1).
- To factor the denominator, we used the same difference of squares formula, but in this case, a is 2a and b is 4, resulting in (2a+4)(2a-4).
- The cancellation of (2-a) and (2a-1) was done because they are opposites and result in -1.
I hope this explanation helps you understand each step clearly.