What is the fourth root of 1?

Understand the Problem

The question is asking for the fourth root of 1, which is a basic concept in mathematics where we are looking for a number that, when raised to the power of four, equals 1.

Answer

The fourth roots of 1 are $1$, $-1$, $i$, and $-i$.

Steps to Solve

Identify the equation We want to find the number (x) such that (x^4 = 1).
Solve for the variable To find the values of (x), we can rewrite the equation: $$ x^4 - 1 = 0 $$ This indicates that (x^4) is equal to 1.
Factor the expression This can be factored by recognizing that it is a difference of squares: $$ (x^2 - 1)(x^2 + 1) = 0 $$
Find possible solutions Now, we set each factor to zero:

For (x^2 - 1 = 0): $$ x^2 = 1 \implies x = 1 \text{ or } x = -1 $$
For (x^2 + 1 = 0): $$ x^2 = -1 \implies x = i \text{ or } x = -i $$ where (i) is the imaginary unit.

List all roots Thus, the fourth roots of 1 are: $$ x = 1, -1, i, -i $$

The fourth roots of 1 are (1), (-1), (i), and (-i).

More Information

The fourth roots of any number can be complex numbers, and the fourth roots of 1 include both real and imaginary numbers. In this case, 1 and -1 are real roots, while (i) and (-i) are imaginary roots.

Tips

Confusing the concept of roots with exponent rules can lead to errors, especially with negative numbers and imaginary units.
Not considering complex roots when dealing with equations like (x^4 = 1).

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