Let's solve the equation [tex]\(5^{x^2 + x} = 1\)[/tex].
### Step-by-Step Solution:
1. Understand the properties of exponents.
- Any non-zero number raised to the power of 0 is equal to 1. This means that if we have [tex]\( a^b = 1 \)[/tex] and [tex]\(a \neq 0\)[/tex], then [tex]\( b \)[/tex] must be 0.
2. Apply this property to our equation.
- Given [tex]\(5^{x^2 + x} = 1\)[/tex], we can say [tex]\(x^2 + x\)[/tex] must be 0 because 5 raised to the power of 0 is 1.
3. Set up the equation.
- From the above reasoning, we set up the equation [tex]\(x^2 + x = 0\)[/tex].
4. Solve the quadratic equation.
- We now have a quadratic equation [tex]\(x^2 + x = 0\)[/tex].
5. Factorize the equation.
- The equation can be factorized as [tex]\(x(x + 1) = 0\)[/tex].
6. Set each factor to zero and solve for [tex]\(x\)[/tex].
- The solutions to the equation [tex]\(x(x + 1) = 0\)[/tex] are obtained by setting each factor equal to zero:
- [tex]\(x = 0\)[/tex]
- [tex]\(x + 1 = 0 \Rightarrow x = -1\)[/tex]
### Solution:
- Therefore, the solutions to the equation [tex]\(5^{x^2 + x} = 1\)[/tex] are [tex]\(x = 0\)[/tex] and [tex]\(x = -1\)[/tex].
In conclusion, the values of [tex]\(x\)[/tex] that satisfy the equation [tex]\(5^{x^2 + x} = 1\)[/tex] are [tex]\(\boxed{0 \text{ and } -1}\)[/tex].
