mmmmmmmmaaaaaaa
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I need help for part b for the two questions below. I have completed part a for both
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Regarding Question 6, you have proven that )
is irrational.
Assume, for a contradiction, that this is not the case, and=\frac{m}{n})
for some 
and 
. Since \approx 1.46>0)
, we see that 
. Thus,
=m)
}=3^m)
}\right)^n=3^m)
Since the number on the left-hand side is divisible by 5, while the number on the right-hand side is not divisible by 5, we reach a contradiction.
Now, since is irrational, it follows that:
=\log _3\left(3\right)+\log _3\left(5\right)=\log _3\left(15\right))
is irrational.
I hope this helps! 
Assume, for a contradiction, that this is not the case, and
Now, since
Paradoxica
-insert title here-
The first part of 7 is a standard exercise which can easily be done using contradiction (assuming you have seen the algebraic bash proof that √2 is irrational)
The second part: If x is irrational, then 2x is irrational (is what we want to show here, for a specific value of x)
Taking the contrapositive: If 2x is rational, then x is rational. This can be proven directly by writing 2x as a quotient of two integers.
In classical logic, the contrapositive of a statement is equivalent to the original statement.
Hence, the original statement is true, and 2√11 = √44 is irrational.
The second part: If x is irrational, then 2x is irrational (is what we want to show here, for a specific value of x)
Taking the contrapositive: If 2x is rational, then x is rational. This can be proven directly by writing 2x as a quotient of two integers.
In classical logic, the contrapositive of a statement is equivalent to the original statement.
Hence, the original statement is true, and 2√11 = √44 is irrational.
mmmmmmmmaaaaaaa
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Thank you for the help. Much appreciatedRegarding Question 6, you have proven that
Assume, for a contradiction, that this is not the case, and
Since the number on the left-hand side is divisible by 5, while the number on the right-hand side is not divisible by 5, we reach a contradiction.
Now, since
I hope this helps!