n^2-4/n^2-2n * n/n^2+10n+16
simplify and state the domain
I am going to assume that you mean
[ (n^2-4)/(n^2-2n) ] * n/(n^2+10n+16 )
Immediately you can see that n = 0 and n = 2 are excluded from the domain, also if
n^2+10 n + 16 = 0 which is if n = -2 or -8
now
[(n-2)(n+2)/n(n-2)] *n /[(n+2)(n+8)]
= 1/(n+8)
your expression definitely needs brackets to make sense
(n^2-4)/(n^2-2n) * n/(n^2+10n+16)
= (n+2)(n-2)/(n(n-2) ) * n( (n+2)(n+5)
= 1/(n+5) , n ≠ -2 , 0, 2 , -5
arggh, go with Damon's factoring of the last part
Whatever made me think that 2x5 = 16 ??????
It's clearly 2 x 8
To simplify the expression (n^2 - 4)/(n^2 - 2n) * n/(n^2 + 10n + 16), we can start by factoring the numerator and denominator of each fraction.
Factorizing the numerator of the first fraction, n^2 - 4, we get (n - 2)(n + 2).
Factorizing the denominator of the first fraction, n^2 - 2n, we get n(n - 2).
The first fraction simplifies to (n - 2)(n + 2)/(n(n - 2)).
The second fraction, n/(n^2 + 10n + 16), cannot be simplified further.
Now, let's multiply the two fractions together:
[(n - 2)(n + 2)/(n(n - 2))] * [n/(n^2 + 10n + 16)]
After cancelling out the common factors (n - 2) in the numerator and denominator, we are left with:
(n + 2)/n * n/(n^2 + 10n + 16)
The 'n' in the numerator and denominator cancels out, leaving us with:
(n + 2)/(n^2 + 10n + 16)
Now, let's determine the domain of the simplified expression. The domain refers to the set of values that n can take without causing any division by zero or undefined expressions.
In this case, we have a polynomial expression in the denominator, n^2 + 10n + 16. To find the domain, we need to ensure that the denominator is not equal to zero. If it were zero, the expression would be undefined.
To find the values of n that make the denominator zero, we need to solve the equation:
n^2 + 10n + 16 = 0
By factoring or using the quadratic formula, we find that the solution to this equation is:
n = -2 or n = -8
Therefore, the domain of the simplified expression is all real numbers except for n = -2 and n = -8. In set notation, the domain can be written as:
Domain: (-∞, -8) U (-8, -2) U (-2, ∞)