For statement 1, start off by raising everything to the second power:
x^2 + y^2 > z^2 --- raise to second power to get:
x^4 + y^4 + 2(x^2)(y^2) > z^4 --- you can't algebraically demonstrate that x^4 + y^4 > z^4, since you've got that 2(x^2)(y^2) over there. That amount is a positive one (since it's the product of three positive numbers), so you don't really know if x^4 + y^4 is indeed greater than z^4 or if it needs the 2(x^2)(y^2) to make it bigger.
I actually tried going against my better judgement and tried to number pick my way through it, but it proved much harder than I thought. But here are two examples:
a. x = 4, y = 0 and z = 2. In this case, we do have that x^2 + y^2 > z^2 and also x^4 + y^4 > z^4.
b. This example was harder to build. I initially thought of fractions, but I realized that it was a dead end. For z, you need something that gets bigger faster after the second power. I eventually picked x = 5, y = 4 and z = 6.
x^2 = 25
y^2 = 16
z^2 = 36
We of course respect the fact that x^2 + y^2 > z^2, since 41 > 36.
BUT:
x^4 = 625
y^4 = 256
z^4 = 216*6, which is greater than 1000.
In this case, you get that x^4 + y^4 < z^4, since x^4 + y^4 < 1000, while z^4 > 1000.
As you can see, this strategy of picking numbers takes two things:
- good knowledge of powers
- patience
This is why for me at least the algebraic method is a lot more comfortable.
