Solution:

Given, (25)^150 = (25x^2)^50

Let's simplify the left-hand side of the equation first.

(25)^150 = (5^2)^150
= 5^(2 x 150)
= 5^300

Now, let's simplify the right-hand side of the equation.

(25x^2)^50 = (5^2 x x^2)^50
= 5^(2 x 50) x x^(2 x 50)
= 5^100 x x^100

Equating the left-hand side and right-hand side, we get:

5^300 = 5^100 x x^100

Dividing both sides by 5^100, we get:

5^(300 - 100) = x^100

5^200 = x^100

Taking the square root of both sides, we get:

x = 5^2^(200/100)

x = 5^2^2

x = 5^4

x = 625

Therefore, the value of x is 625.

Explanation:

We are given an equation involving exponents, and we need to solve for the value of x. To do this, we need to simplify both sides of the equation and then equate them. We then isolate x on one side of the equation by performing the necessary mathematical operations.

In this case, we simplified the left-hand side of the equation using the laws of exponents, and we simplified the right-hand side of the equation by expanding (25x^2)^50 using the laws of exponents.

We then equated the left-hand side and right-hand side of the equation and simplified to get x^100 on one side of the equation. We then took the square root of both sides to isolate x and obtained the value of x as 625.

Overall, this problem involves a combination of algebraic manipulation and the use of laws of exponents to solve for an unknown variable.