Determine Why Hyp12/3.7.5 Is Irrational

#9: Irrational or Not:
Problem: Determine if each of the following numbers is rational or irrational.
√16/20, √12/7.5, -147, 0, √3/3 Solution:
√16/20 is irrational because its decimal is non-repeating and never ending.
√12/7.5 is irrational because its decimal is non-repeating and never ending.
147 is rational because it can become a fraction
0 is rational because it can be written as a fraction
√3/3 is irrational because if you take the square root of a number that is not a square it is going to be irrational. If you take an irrational number and divide it by any other number its still going to be irrational so √3/3 is irrational.

#12: Irrational:
Problem:Show that the Value given is irrational
√7
If it is rational we would say that √7 = a/b where a and b are in the lowest terms and b ≠ 0. If a and b are in lowest term, it means they Do Not share any common factors.
Square both sides - 7 = a2/b2
Multiply both sides by b2 - 7b2 = a2
Since both sides are equal, if 7 divides the left hand side it must also divide the right hand side, so we can rewrite a as 7k, where k is some integer ≠ 0.
7b2 = (7k)2 which is 7b2 = 49k2 which is b2 = 7k2 a and b are divisible by 7, which means they do share a common factor of 7. But the rules state that if a and b are in lowest terms, they do not share any common factor. So, we have a contradiction which means that out assumption (√7 is rational) is false. Since we know that √7 is not rational we prove that √7 is irrational.

#18: Another Irrational exponent:
Problem: Suppose the E is the number such that 13E = 8. Show that E is an irrational number.

Solution:
E is irrational – proof by