Problem 1 - Entrance Test

To solve this problem, we can start by trying different values of x to see if we can find one that works. We can also use algebraic manipulations to simplify the equation. For example, we can divide both sides of the equation by 10^x to get (3/10)^x + (7/10)^x = 1. We can then try to find a value of x that satisfies this equation. After some trial and error, we find that x = 3 satisfies the equation, since (3/10)^3 + (7/10)^3 = 27/1000 + 343/1000 = 370/1000 = 0.37, which is close to, but not quite equal to 1. However, if we try x = 4, we get (3/10)^4 + (7/10)^4 = 81/10000 + 2401/10000 = 2482/10000 = 0.2482, which is less than 1. On the other hand, if we try x = 2, we get (3/10)^2 + (7/10)^2 = 9/100 + 49/100 = 58/100 = 0.58, which is also less than 1. Therefore, the value of x that satisfies the equation is x = 3.