Understanding Exponent Rules: (x^4)^2 = x^12 / x^5
This article explores the simplification of the equation (x^4)^2 = x^12 / x^5 using fundamental exponent rules.
Understanding the Exponent Rules
Before diving into the simplification, let's revisit two key exponent rules:

Power of a power: (x^m)^n = x^(m*n)
 This rule states that raising a power to another power is equivalent to multiplying the exponents.

Division of powers with the same base: x^m / x^n = x^(mn)
 This rule states that dividing powers with the same base is equivalent to subtracting the exponents.
Simplifying the Equation
Now, let's apply these rules to simplify our equation:

Simplify the left side: (x^4)^2 = x^(4*2) = x^8
 We applied the "power of a power" rule.

Simplify the right side: x^12 / x^5 = x^(125) = x^7
 We applied the "division of powers with the same base" rule.
Verifying the Equation
Therefore, the simplified equation becomes x^8 = x^7. This equation is not true for all values of x, and it's important to understand why.
For example:
 If x = 2, then x^8 = 256 and x^7 = 128.
 If x = 0, then both sides are equal to 0.
The equation holds true only for the specific case where x = 1.
Conclusion
Simplifying the equation (x^4)^2 = x^12 / x^5 using exponent rules leads to x^8 = x^7. This equation highlights the importance of understanding exponent rules for accurate simplification. Furthermore, it demonstrates that applying these rules may not always lead to a universally true equation. It's essential to remember that the equation holds true only for specific values of x.