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 \(-i\).
Answer for screen readers
The fourth roots of 1 are (1), (-1), (i), and (-i).
Steps to Solve
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Identify the equation We want to find the number (x) such that (x^4 = 1).
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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.
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Factor the expression This can be factored by recognizing that it is a difference of squares: $$ (x^2 - 1)(x^2 + 1) = 0 $$
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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).
