To determine if the given equation [tex]\( x^4 = 256 \)[/tex] is an exponential equation, we need to understand the characteristics of exponential equations and how they differ from other types of equations.
### Characteristics of Exponential Equations
An exponential equation is one where a variable appears in the exponent. It generally takes the form [tex]\( a \cdot b^x = c \)[/tex], where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are constants, [tex]\( b \)[/tex] is a positive real number not equal to 1, [tex]\( x \)[/tex] is the variable, and [tex]\( c \)[/tex] is another constant.
### Analysis of the Given Equation
Consider the equation [tex]\( x^4 = 256 \)[/tex]:
1. Form of the Equation: The given equation expresses [tex]\( x \)[/tex] raised to the power of 4, set equal to 256. In an exponential equation, the variable [tex]\( x \)[/tex] should be in the exponent, but in this equation, [tex]\( x \)[/tex] is the base and 4 is the exponent.
2. Polynomial Nature: The equation [tex]\( x^4 = 256 \)[/tex] is actually a polynomial equation. Specifically, it is a fourth-degree polynomial, since the highest power of the variable [tex]\( x \)[/tex] is 4.
3. Absence of an Exponential Term: In an exponential equation, such as [tex]\( 2^x = 16 \)[/tex], the variable [tex]\( x \)[/tex] is in the exponent. The given equation does not match this form; it lacks the structure [tex]\( b^x \)[/tex] where [tex]\( b \)[/tex] is a constant base and [tex]\( x \)[/tex] is the exponent.
### Conclusion
The given equation [tex]\( x^4 = 256 \)[/tex] is not an exponential equation because the variable [tex]\( x \)[/tex] is the base raised to a power (4), rather than the exponent itself. Instead, this equation is a polynomial equation of the fourth degree.
