Sum/difference:
Let
This means that
Now, assume that 
is rational. The sum/difference of two rational numbers is still rational (so 5-x is rational), and the division by 3 doesn't change this. So, you have that the square root of 8 equals a rational number, which is false. The mistake must have been supposing that was rational, which proves that the sum/difference of the two given terms was irrational
Multiplication/division:
The logic is actually the same: if we multiply the two terms we get
if again we assume x to be rational, we have
But if x is rational, so is -x/15, and again we come to a contradiction: we have the square root of 8 on one side, which is irrational, and -x/15 on the other, which is rational. So, again, x must have been irrational. You can prove the same claim for the division in a totally similar fashion.
Hi there!
Since the formula for finding the perimeter of a rectangle is length×2 + width × 2, we can follow this rule.
2 (4x + 5) + 2 (8x - 4)
= 8x + 10 + 16x - 8
= 24x + 2
So, the perimeter is 24x + 2.
Hope this helps!
