Simplify the sum. State any restrictions on the variable. Show your work.
5v+8/v-3 + 8x+8/v-3
To simplify the sum, we can combine the two fractions by finding a common denominator.
The common denominator for both fractions is (v-3).
We can rewrite the sum as:
((5v+8) + (8x+8))/(v-3)
Simplifying further, we have:
(5v + 8 + 8x + 8)/(v-3)
Combining like terms gives us:
(5v + 8x + 16)/(v-3)
So the simplified sum is (5v + 8x + 16)/(v-3).
As for restrictions on the variable:
The variable v cannot equal 3, as that would make the denominator (v-3) equal to zero, causing division by zero which is undefined.
To simplify the sum (5v + 8)/(v - 3) + (8x + 8)/(v - 3), we can combine the two fractions since they have the same denominator.
Step 1: Combine the fractions
[(5v + 8) + (8x + 8)] / (v - 3)
Step 2: Simplify the numerator
Combine like terms in the numerator:
(5v + 8x + 8v + 8) / (v - 3)
Step 3: Combine like terms in the numerator
(13v + 8x + 8) / (v - 3)
So, the simplified expression is (13v + 8x + 8) / (v - 3).
Restrictions on the variable:
The only restriction is that the denominator (v - 3) cannot be zero since division by zero is undefined. Therefore, v ≠ 3.
To simplify the sum (5v + 8/v - 3) + (8x + 8/v - 3), we need to combine like terms.
First, let's focus on the terms involving v. We have 5v and 8/v, so the first step is finding a common denominator. The common denominator for v and 1/v is v.
To get the common denominator, we multiply the numerator and denominator of 8/v by v: (8v/v^2).
Now, we can rewrite the expression as:
(5v + 8v/v^2 - 3) + (8x + 8/v - 3)
Next, let's combine the v terms:
5v + 8v/v^2 = (5v^3 + 8v)/v^2
Rewriting the sum now, we have:
(5v^3 + 8v)/v^2 - 3 + 8x + 8/v - 3
Next, we combine the x terms:
(5v^3 + 8v)/v^2 - 3 + 8x + 8/v - 3 = (5v^3 + 8v)/v^2 + 8x + 8/v - 6
This is the simplified form of the sum.
Restrictions on the variable:
There are no specific restrictions mentioned in the problem. However, it is possible that the expression is undefined for certain values of v if they result in a zero denominator (v^2 = 0 or v = 0). So, the restriction on the variable would be that v should not be equal to zero.