- The given equation is 2^(x^2) = 3^(y^2) = 12^(z^2).
- This equation represents a relationship between three different exponential expressions where the bases are 2, 3, and 12 respectively.



- Let's break down the equation and analyze each part separately:



- Since the two exponential expressions are equal, their exponents must also be equal.
- This gives us x^2 = y^2.



- Similarly, since the two exponential expressions are equal, their exponents are also equal.
- This gives us y^2 = z^2.



- Now, we have two equations: x^2 = y^2 and y^2 = z^2.
- By transitive property, we can conclude that x^2 = z^2.
- Therefore, x = z.



- In conclusion, the relationship between the exponents x, y, and z in the given equation is x = z.
- This relationship helps us understand the connection between the bases 2, 3, and 12 in the exponential expressions.
- By analyzing the equation and solving for the exponents, we can better comprehend the underlying patterns and connections within the mathematical expression.