I have no idea how you would do this problem: Express this expression as a rational number in lowest terms.
(1000!-999!-998!)/(1000!+999!+998!)
(1000!-999!-998!)/(1000!+999!+998!)=
998!(1000*999-999-1)/998!(1000*999+999+1)=
(1000*999-1000)/(1000*999+1000)=
(1000*998)/(1000*1000)=998/1000 and u can simplify this number
You have to realize that factorials can be written in several ways
e.g.
12! = 12*11*10*9! or
12! = 12*10*10! I am saying you can stop anywhere
(1000!-999!-998!)/(1000!+999!+998!)
= (1000*999*998!-999*998!-998!)/(1000*999*998!+999*998!+998!)
=998!(1000*999-999-1)/[998!(1000*999+999+1)]
= 998000/1000000
=499/500
Both numerator and denominator have a common factor of 998!. Cancel that out and you have
[1000*999 -999-1]/[1000*999 + 999+1]
=[1000(999-1)]/[1000(999+1)]
=449/500
499/500
To express the given expression as a rational number in lowest terms, we need to simplify the numerator and denominator separately and then divide them.
First, let's simplify the numerator:
Numerator = 1000! - 999! - 998!
Notice that all the terms have a common factor of 998! (the factorial of 998). By factoring out 998!, we can simplify the numerator as follows:
Numerator = 998!(1000*999 - 999 - 1)
= 998!(999,000 - 999 - 1)
= 998!(998,000)
Next, let's simplify the denominator:
Denominator = 1000! + 999! + 998!
Similar to the numerator, we can factor out 998! from all the terms in the denominator:
Denominator = 998!(1000 + 999 + 1)
= 998!(2,000)
Now, let's divide the simplified numerator by the simplified denominator to find the rational number in lowest terms:
Rational number = Numerator / Denominator
= (998!(998,000)) / (998!(2,000))
Since the 998! factor cancels out in the numerator and denominator, we can simplify the expression further:
Rational number = (998,000) / (2,000)
= 499
Therefore, the given expression can be expressed as the rational number 499 in its lowest terms.