Final answer:
The given statement "Is any real number exactly 1 less than its fourth power" is false.
Step-by-step explanation:
To answer this question, we first need to understand what a fourth power is. A fourth power is a number raised to the power of 4, also known as a quartic power. For example, if we have the number 2, its fourth power would be 2^4, which is equal to 16.
Now, let's assume that there exists a real number x that is exactly 1 less than its fourth power. In mathematical terms, this would mean that x = x^4 - 1. To determine if this is true or false, we need to solve for x.
x^4 - 1 = x
x^4 - x - 1 = 0
This is now a quartic equation, which can be solved using various methods such as factoring, completing the square, or using the quadratic formula. However, none of these methods will give us a real solution for x. In fact, the solutions to this equation are complex numbers, which means that this statement is false.
But why is this statement false? Let's take a closer look at the equation x = x^4 - 1. This equation can also be written as x^4 - x - 1 = 0, which is a quartic equation. In general, quartic equations have at most 4 solutions. However, in this case, we have 4 complex solutions. This means that there is no real number that satisfies this equation, making the statement false.
In conclusion, there is no real number that is exactly 1 less than its fourth power. This can be proven by solving the equation x = x^4 - 1, which results in complex solutions. Therefore, the answer to the given question is false.