Final answer:
Given that x raised to any positive power multiplied by x raised to the negative of that power should equal 1, the expression x¹⁹ should be equal to 1 regardless of the value of x.
Step-by-step explanation:
If x * x⁻¹ = √2, then by definition x⁻¹ is the multiplicative inverse of x, which means that x * x⁻¹ = 1. The equation given suggests that x⁻¹ is not the multiplicative inverse in this context, as it gives a different product. Therefore, the equation must be a special condition or context applied to x and its inverse.
However, if we look at the exponents and the laws of exponents, in any conventional sense, x * x⁻¹ should indeed equal 1, given that x to any positive exponent times x to the corresponding negative exponent cancels out to 1. So for the expression x²⁰, which represents x raised to the 20th power, it doesn't depend on x since any number raised to the 0 power is 1. Hence, option b) x²⁰ = 1 should be the correct answer.