Question Analysis:
To solve the given equations, we need to find the values of x and y that satisfy both equations simultaneously. The equations involve exponents and roots, so we will use properties of exponents and roots to simplify and solve them.

Solution:

Equation 1:
Given: √32^x ÷ 2^y = 1
To simplify this equation:
√32^x = 2^y
(32^x)^(1/2) = 2^y
2^(5x/2) = 2^y
Now, equating the exponents:
5x/2 = y
5x = 2y
x = 2y/5

Equation 2:
Given: 16^(4-x/2) - 8y = 0
To simplify this equation:
16^(4-x/2) = 8y
(2^4)^(4-x/2) = 8y
2^(4*(4-x/2)) = 8y
2^(16-2x) = 8y
2^16 * 2^(-2x) = 8y
2^16 / 2^(2x) = 8y
2^(16-2x) = 8y
Now, we can substitute the value of x from Equation 1 into Equation 2:
2^(16 - 2(2y/5)) = 8y
2^(16 - 4y/5) = 8y
2^(80/5 - 4y/5) = 8y
2^(76/5 - 4y/5) = 8y
2^(76-4y)/5 = 8y
2^(76-4y) = 8y^5
At this point, we have a system of equations that can be solved simultaneously to find the values of x and y that satisfy both equations.