The problem asks for the product of the roots of the polynomial equation $x^4 + 1 = 0$.

1. General Rule for Product of Roots

For a general polynomial equation of degree $n$:

$$a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 = 0$$

The product of the roots is given by the formula:

$$\text{Product of roots} = (-1)^n \frac{a_0}{a_n}$$

where $a_0$ is the constant term and $a_n$ is the coefficient of the highest power term ($x^n$).

2. Apply to the Given Equation

The equation is $x^4 + 1 = 0$. This can be written as:

$$1 \cdot x^4 + 0 \cdot x^3 + 0 \cdot x^2 + 0 \cdot x + 1 = 0$$

• Degree of the polynomial: $n = 4$
• Coefficient of the highest power term ($x^4$): $a_4 = 1$
• Constant term ($x^0$): $a_0 = 1$

The product of the roots is:

$$\text{Product} = (-1)^4 \frac{a_0}{a_4}$$ $$\text{Product} = (1) \cdot \frac{1}{1}$$ $$\text{Product} = 1$$

3. Conclusion

The product of the roots of the equation $x^4 + 1 = 0$ is 1.

The answer (in integer) is 1.