To find the roots of the expression [tex]\( 5^{x^2 - 9x + 17} = 0 \)[/tex], we need to determine the values of [tex]\( x \)[/tex] that will make the expression equal to zero. Let's analyze the situation step-by-step.
1. Understand the Nature of the Expression:
The given expression is [tex]\( 5^{x^2 - 9x + 17} \)[/tex].
2. Exponential Function Properties:
Note that an exponential function with a positive base (in this case, 5) never equals zero. This is because any positive number raised to any real power will always result in a positive number.
3. Logical Conclusion:
Since [tex]\( 5^{y} > 0 \)[/tex] for any real number [tex]\( y \)[/tex], [tex]\( 5^{x^2 - 9x + 17} \)[/tex] cannot be zero for any real value of [tex]\( x \)[/tex].
4. Confirm the Exponent:
While the exponent [tex]\( x^2 - 9x + 17 \)[/tex] is a quadratic expression and can have real values (positive, negative, or zero), this fact does not change the nature of the exponential function. As long as the base is 5 (a positive number), raising it to any real power cannot give a result of zero.
Therefore, there are no real roots [tex]\( x \)[/tex] that satisfy the equation [tex]\( 5^{x^2 - 9x + 17} = 0 \)[/tex]. The expression cannot equal zero for any real value of [tex]\( x \)[/tex].
