Answer:
a)
We know that:
a, b > 0
a < b
With this, we want to prove that a^2 < b^2
Well, we start with:
a < b
If we multiply both sides by a, we get:
a*a < b*a
a^2 < b*a
now let's go back to the initial inequality.
a < b
if we now multiply both sides by b, we get:
a*b < b*b
a*b < b^2
Then we have the two inequalities:
a^2 < b*a
a*b < b^2
a*b = b*a
Then we can rewrite this as:
a^2 < b*a < b^2
This means that:
a^2 < b^2
b) Now we know that a.b > 0, and a^2 < b^2
With this, we want to prove that a < b
So let's start with:
a^2 < b^2
only with this, we can know that a*b will be between these two numbers.
Then:
a^2 < a*b < b^2
Now just divide all the sides by a or b.
if we divide all of them by a, we get:
a^2/a < a*b/a < b^2/a
a < b < b^2/a
In the first part, we have a < b, this is what we wanted to get.
Another way can be:
a^2 < b^2
divide both sides by a^2
1 < b^2/a^2
Let's apply the square root in both sides:
√1 < √( b^2/a^2)
1 < b/a
Now we multiply both sides by a:
a < b