Given x*y=xy-2x-2y+6, x*2=2, calculate -2009*-2008*-----*-1*0*1*----2009?

Expert Answers

An illustration of the letter 'A' in a speech bubbles

You should use the information provided by the problem `x*2 = 2` , hence, you may group the terms around the term 2 such that:

`((-2009)*......*(-1)*0*1)*2*(3*4*...*2009)`

Reasoning by analogy yields that `((-2009)*......*(-1)*0*1)*2 = 2.`

You should prove that the given law of composition is associative, hence, it satisfies the associative law such that:

`(x*y)*z= x*(x*z)`

You need to use the following form of the law of composition, such that:

`x*y = (x-2)(y-2) + 2`

`(x*y)*z = ((x-2)(y-2) + 2)*z = ((x-2)(y-2) + 2 - 2)(z-2) + 2`

`(x*y)*z = (x-2)(y-2)(z-2) + 2`

`y*z = (y-2)(z-2) + 2`

`x*(y*z) = x*((y-2)(z-2) + 2)`

`x*(y*z) = (x-2)((y-2)(z-2) + 2 - 2) + 2`

You need to evaluate if `(x*y)*z = x*(y*z)`  such that:

`(x-2)(y-2)(z-2) + 2 = (x-2)((y-2)(z-2)) + 2`

Reducing 2 both sides yields:

`(x-2)(y-2)(z-2) = (x-2)(y-2)(z-2)`

Since the law is associative and commutative, hence `x*2 = 2*x = 2.`

Reasoning by analogy yields:

`2*(3*4*...*2009) = 2`

Hence, evaluating the expression `((-2009)*......*(-1)*0*1)*2*(3*4*...*2009), ` using the information provided by the problem, yields `((-2009)*......*(-1)*0*1)*2*(3*4*...*2009) = 2.`

See eNotes Ad-Free

Start your 48-hour free trial to get access to more than 30,000 additional guides and more than 350,000 Homework Help questions answered by our experts.

Get 48 Hours Free Access
Approved by eNotes Editorial